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1
+ ---
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+ license: apache-2.0
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+ datasets:
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+ - TeichAI/claude-4.5-opus-high-reasoning-250x
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+ base_model:
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+ - allura-forge/Llama-3.3-8B-Instruct
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+ language:
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+ - en
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+ - fr
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+ - de
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+ - es
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+ - it
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+ - pt
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+ - zh
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+ - ja
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+ - ru
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+ - ko
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+ tags:
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+ - thinking
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+ - reasoning
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+ - instruct
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+ - Claude4.5-Opus
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+ - creative
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+ - creative writing
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+ - fiction writing
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+ - plot generation
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+ - sub-plot generation
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+ - fiction writing
29
+ - story generation
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+ - scene continue
31
+ - storytelling
32
+ - fiction story
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+ - science fiction
34
+ - romance
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+ - all genres
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+ - story
37
+ - writing
38
+ - vivid prosing
39
+ - vivid writing
40
+ - fiction
41
+ - roleplaying
42
+ - bfloat16
43
+ - role play
44
+ - 128k context
45
+ - llama3.3
46
+ - llama-3
47
+ - llama-3.3
48
+ - unsloth
49
+ - finetune
50
+ pipeline_tag: text-generation
51
+ library_name: transformers
52
+ ---
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+
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+ <H2>Llama3.3-8B-Instruct-Thinking-Claude-4.5-Opus-High-Reasoning</H2>
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+
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+ <img src="matrix-neo-reloaded-fight.gif" style="float:right; width:300px; height:300px; padding:10px;">
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+
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+ What madness is this?
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+
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+ Someone found "Llama3.3-8B" source (never publicly released) in the "wild", then it was adjusted back to 128k and then
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+ I added my own special madness:
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+
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+ Training the model with Unsloth (3 epochs) and Claude 4.5-Opus High Reasoning dataset.
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+
65
+ This has created an Instruct/Thinking hybrid (128k context, Llama 3.3 model).
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+
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+ Note this tuning was only to create an instruct/thinking model, not to update the model's core knowledge / root training.
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+
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+ 1 example at bottom of the page.
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+
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+ <B>HERETIC / Uncensored Version:</B>
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+
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+ https://huggingface.co/DavidAU/Llama3.3-8B-Instruct-Thinking-Heretic-Uncensored-Claude-4.5-Opus-High-Reasoning
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+
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+ <B>Thinking (will activate automatically) prompts - examples: </B>
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+
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+ Explain orbital mechanics including detailed math and examples. (example output at bottom of the page)
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+
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+ Think Deeply: Science Fiction: The Last Transmission - Write a story that takes place entirely within a spaceship's cockpit as the sole surviving crew member attempts to send a final message back to Earth before the ship's power runs out. The story should explore themes of isolation, sacrifice, and the importance of human connection in the face of adversity. If the situation calls for it, have the character(s) curse and swear to further the reader's emotional connection to them. 800-1000 words.
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+
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+ Think deeply: Tell me a horror story.
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+
83
+ (certain phrases/words will automatically activate thinking)
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+
85
+ <B>Instruct (thinking may NOT activate) prompts - examples:</B>
86
+
87
+ Science Fiction: The Last Transmission - Write a story that takes place entirely within a spaceship's cockpit as the sole surviving crew member attempts to send a final message back to Earth before the ship's power runs out. The story should explore themes of isolation, sacrifice, and the importance of human connection in the face of adversity. If the situation calls for it, have the character(s) curse and swear to further the reader's emotional connection to them. 800-1000 words.
88
+
89
+ Tell me a horror story.
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+
91
+ <B>SETTINGS (suggested):</B>
92
+
93
+ Temp .7, rep pen 1.05, topp: .95, minp .05, topk: 40
94
+
95
+ Min context window: 4k, but suggest 8k+.
96
+
97
+ NO system prompt [thinking tags will self generate].
98
+
99
+ Suggest Quant of Q4KS (non imatrix) or IQ3_M (imatrix) or higher ; lower quants may have reasoning issues/activation issues.
100
+
101
+ [more to come]
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+
103
+ ---
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+
105
+ Special thanks to:
106
+
107
+ ---
108
+
109
+ https://huggingface.co/allura-forge/Llama-3.3-8B-Instruct
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+ (found the model!)
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+
112
+ https://huggingface.co/shb777/Llama-3.3-8B-Instruct-128K
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+ (adjusted to 128k, and other fixes)
114
+
115
+ https://huggingface.co/datasets/TeichAI/claude-4.5-opus-high-reasoning-250x
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+ (for the F..ing amazing dataset)
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+
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+ and Unsloth for making tuning too easy:
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+
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+ https://github.com/unslothai/unsloth
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+
122
+ Details on the "madness":
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+
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+ https://www.reddit.com/r/LocalLLaMA/comments/1pz7bmv/llama338binstruct/
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+
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+ https://www.reddit.com/r/LocalLLaMA/comments/1q06ddc/update_on_the_llama_33_8b_situation/
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+
128
+ ---
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+
130
+ <B>Settings: CHAT / ROLEPLAY and/or SMOOTHER operation of this model:</B>
131
+
132
+ In "KoboldCpp" or "oobabooga/text-generation-webui" or "Silly Tavern" ;
133
+
134
+ Set the "Smoothing_factor" to 1.5
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+
136
+ : in KoboldCpp -> Settings->Samplers->Advanced-> "Smooth_F"
137
+
138
+ : in text-generation-webui -> parameters -> lower right.
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+
140
+ : In Silly Tavern this is called: "Smoothing"
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+
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+
143
+ NOTE: For "text-generation-webui"
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+
145
+ -> if using GGUFs you need to use "llama_HF" (which involves downloading some config files from the SOURCE version of this model)
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+
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+ Source versions (and config files) of my models are here:
148
+
149
+ https://huggingface.co/collections/DavidAU/d-au-source-files-for-gguf-exl2-awq-gptq-hqq-etc-etc-66b55cb8ba25f914cbf210be
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+
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+ OTHER OPTIONS:
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+
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+ - Increase rep pen to 1.1 to 1.15 (you don't need to do this if you use "smoothing_factor")
154
+
155
+ - If the interface/program you are using to run AI MODELS supports "Quadratic Sampling" ("smoothing") just make the adjustment as noted.
156
+
157
+ <B>Highest Quality Settings / Optimal Operation Guide / Parameters and Samplers</B>
158
+
159
+ This a "Class 1" model:
160
+
161
+ For all settings used for this model (including specifics for its "class"), including example generation(s) and for advanced settings guide (which many times addresses any model issue(s)), including methods to improve model performance for all use case(s) as well as chat, roleplay and other use case(s) please see:
162
+
163
+ [ https://huggingface.co/DavidAU/Maximizing-Model-Performance-All-Quants-Types-And-Full-Precision-by-Samplers_Parameters ]
164
+
165
+ You can see all parameters used for generation, in addition to advanced parameters and samplers to get the most out of this model here:
166
+
167
+ [ https://huggingface.co/DavidAU/Maximizing-Model-Performance-All-Quants-Types-And-Full-Precision-by-Samplers_Parameters ]
168
+
169
+ ---
170
+
171
+ Example: Q4KS , Lmstudio, "suggested settings"
172
+
173
+ NOTE: Some loss of formatting.
174
+
175
+ ---
176
+
177
+ PROMPT:
178
+
179
+ <B>
180
+ Explain orbital mechanics including detailed math and examples.
181
+ </B>
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+
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+ Thinking/OUTPUT:
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+
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+ <style type="text/css">
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+ @page { size: 21cm 29.7cm; margin: 2cm }
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+ p { line-height: 115%; margin-bottom: 0.25cm; background: transparent }
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+ h2 { margin-top: 0.35cm; margin-bottom: 0.21cm; background: transparent; page-break-after: avoid }
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+ h2.western { font-family: "Liberation Serif", serif; font-weight: bold; font-size: 18pt }
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+ h2.cjk { font-size: 18pt; font-family: "NSimSun"; font-weight: bold }
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+ h2.ctl { font-family: "Lucida Sans"; font-size: 18pt; font-weight: bold }
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+ h3 { margin-top: 0.25cm; margin-bottom: 0.21cm; background: transparent; page-break-after: avoid }
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+ h3.western { font-family: "Liberation Serif", serif; font-weight: bold; font-size: 14pt }
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+ h3.cjk { font-size: 14pt; font-family: "NSimSun"; font-weight: bold }
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+ h3.ctl { font-family: "Lucida Sans"; font-size: 14pt; font-weight: bold }
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+ h1 { margin-bottom: 0.21cm; background: transparent; page-break-after: avoid }
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+ h1.western { font-family: "Liberation Serif", serif; font-weight: bold; font-size: 24pt }
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+ h1.cjk { font-size: 24pt; font-family: "NSimSun"; font-weight: bold }
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+ h1.ctl { font-family: "Lucida Sans"; font-size: 24pt; font-weight: bold }
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+ h4 { margin-top: 0.21cm; margin-bottom: 0.21cm; background: transparent; page-break-after: avoid }
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+ h4.western { font-family: "Liberation Serif", serif; font-weight: bold; font-size: 12pt }
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+ h4.cjk { font-size: 12pt; font-family: "NSimSun"; font-weight: bold }
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+ h4.ctl { font-family: "Lucida Sans"; font-size: 12pt; font-weight: bold }
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+ td p { orphans: 0; widows: 0; background: transparent }
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+ th p { font-weight: bold; orphans: 0; text-align: center; widows: 0; background: transparent }
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+ pre { background: transparent }
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+ pre.western { font-family: "Liberation Mono", monospace; font-size: 10pt }
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+ pre.cjk { font-size: 10pt; font-family: "NSimSun", monospace }
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+ pre.ctl { font-family: "Liberation Mono", monospace; font-size: 10pt }
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+ strong { font-weight: bold }
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+ code.western { font-family: "Liberation Mono", monospace }
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+ code.cjk { font-family: "NSimSun", monospace }
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+ code.ctl { font-family: "Liberation Mono", monospace }
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+ </style>
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+
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+ <span style="display: inline-block; border: none; padding: 0cm">llama-3.3-8b-instruct-claude-4.5-opus-high-reasoning-checkpoint-90-q4_k_s.gguf</span></p>
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+ <p style="border: none; padding: 0cm"><br/>
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+ <br/>
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+
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+ </p>
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+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><font color="#ff4000"><b>Thought
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+ for 19.49 seconds</b></font></span></p>
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+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">This
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+ is a comprehensive request about orbital mechanics. The person wants
225
+ me to explain the subject in detail, including mathematical
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+ derivations and examples. Let me organize this thoroughly.</span></p>
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+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">I'll
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+ cover:</span></p>
229
+ <ol>
230
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Fundamentals
231
+ of Orbital Mechanics</span></p></li>
232
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Kepler's
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+ Laws</span></p></li>
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+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Elliptical
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+ Orbits</span></p></li>
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+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
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+ Elements</span></p></li>
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+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Vis-Viva
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+ Equation</span></p></li>
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+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-Analytical
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+ Methods</span></p></li>
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+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Numerical
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+ Integration</span></p></li>
244
+ </ol>
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+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
246
+ each section, I'll include mathematical derivations and examples.</span></p>
247
+ <h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
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+ <span style="display: inline-block; border: none; padding: 0cm">1.
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+ Fundamentals of Orbital Mechanics</span></h2>
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+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
251
+ mechanics deals with the motion of objects in space under
252
+ gravitational forces. The two-body problem (one object orbiting
253
+ another) is fundamental.</span></p>
254
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
255
+ quantities:</span></p>
256
+ <ul>
257
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-major
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+ axis (a)</span></p></li>
259
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Eccentricity
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+ (e)</span></p></li>
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+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
262
+ period (T)</span></p></li>
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+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Gravitational
264
+ parameter (μ = GM)</span></p></li>
265
+ </ul>
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+ <h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
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+ <span style="display: inline-block; border: none; padding: 0cm">2.
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+ Kepler's Laws</span></h2>
269
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Kepler's
270
+ laws provide a framework for understanding orbital motion, derived
271
+ from Tycho Brahe's observations and Newton's laws.</span></p>
272
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">First
273
+ Law: Law of Ellipses</span></h3>
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+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">An
275
+ orbit is an ellipse with one focus at the center of mass of the
276
+ system.</span></p>
277
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Second
278
+ Law: Equal Area Rule</span></h3>
279
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
280
+ line connecting the planet to the Sun sweeps out equal areas in equal
281
+ times.</span></p>
282
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Third
283
+ Law: Harmonies of the Spheres</span></h3>
284
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">There
285
+ is a universal relationship between orbital period and semi-major
286
+ axis.</span></p>
287
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Mathematically:
288
+ T² = (4π²/GM) * a³</span></p>
289
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Or
290
+ for Earth around the Sun: T² ≈ 9.95 × 10⁷ * a³</span></p>
291
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
292
+ circular orbits: v²/a = GM/r² = GM/a² = μ/a</span></p>
293
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
294
+ Velocity</span></h3>
295
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
296
+ velocity of an object in orbit depends on its distance from the
297
+ center of gravity.</span></p>
298
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
299
+ circular orbits: v = √(GM/r)</span></p>
300
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
301
+ elliptical orbits (vis-viva equation): v² = GM(2/r - 1/a)</span></p>
302
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
303
+ energy is a fundamental concept in understanding orbital mechanics.
304
+ The total energy (E) of an orbiting body can be expressed as the sum
305
+ of kinetic and potential energy components.</span></p>
306
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
307
+ circular orbits, the total energy is: E = -GM²/2a</span></p>
308
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">This
309
+ reveals the negative sign indicating gravitational binding energy
310
+ that keeps the object in continuous orbit around a celestial body.</span></p>
311
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
312
+ orbital parameters include semi-major axis (a), eccentricity (e),
313
+ inclination (i), longitude of the ascending node (Ω), argument of
314
+ periapsis (ω), and true anomaly (f). These parameters collectively
315
+ define the precise orbit of an object in space.</span></p>
316
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
317
+ elliptical orbits, the shape is determined by the semi-major axis
318
+ (a), eccentricity (e), and focal length (c = ae). The semi-major axis
319
+ represents the average distance between the orbiting body and the
320
+ center of mass, while eccentricity describes the orbital shape
321
+ ranging from 0 (circular) to 1 (parabolic).</span></p>
322
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Eccentricity
323
+ (e) quantifies the deviation from a perfect circle, with values below
324
+ 1.0 indicating elliptical shapes and 1.0 or higher representing
325
+ parabolic trajectories.</span></p>
326
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
327
+ elements are critical parameters that define an object's orbital
328
+ characteristics. These include semi-major axis (a), eccentricity (e),
329
+ inclination (i), longitude of the ascending node (Ω), argument of
330
+ periapsis (ω), and true anomaly (f).</span></p>
331
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-major
332
+ axis (a) represents the average distance from the center of mass to
333
+ the orbiting body, while eccentricity (e) describes the orbital
334
+ shape. Inclination (i) measures the angle between the orbital plane
335
+ and a reference plane, typically the ecliptic.</span></p>
336
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Longitude
337
+ of the ascending node (Ω) identifies the position of the ascending
338
+ node, marking the point where the object crosses from south to north
339
+ in its orbit. Argument of periapsis (ω) specifies the position of
340
+ the closest approach (periapsis) within the orbital plane.</span></p>
341
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">True
342
+ anomaly (f) describes the object's current position along its orbit,
343
+ ranging from 0° at periapsis to 180° at apoapsis and continuing
344
+ back to 0° at the next periapsis passage.</span></p>
345
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Vis-viva
346
+ equation provides a fundamental relationship between orbital
347
+ velocity, gravitational parameter, and radial distance. This equation
348
+ reveals how an object's speed changes with distance from its central
349
+ body, offering insights into orbital mechanics and gravitational
350
+ interactions.</span></p>
351
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
352
+ elliptical orbits, the vis-viva equation extends beyond circular
353
+ motion by incorporating eccentricity (e) and radial distance (r). The
354
+ equation shows that velocity remains constant only at specific points
355
+ (periapsis or apoapsis) in an elliptical trajectory, with varying
356
+ speeds throughout the orbit.</span></p>
357
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Numerical
358
+ integration methods offer precise computational approaches for
359
+ tracking complex orbital trajectories. By discretizing time into
360
+ smaller intervals and applying numerical techniques, scientists can
361
+ accurately model orbital dynamics, even when analytical solutions
362
+ become impractical.</span></p>
363
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
364
+ numerical integration techniques include Runge-Kutta methods, Verlet
365
+ algorithms, and symplectic integrators. These approaches enable
366
+ detailed simulations of celestial body interactions, gravitational
367
+ influences, and orbital perturbations.</span></p>
368
+ <p style="border: none; padding: 0cm"><br/>
369
+ <br/>
370
+
371
+ </p>
372
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><font color="#ff4000"><b>OUTPUT:</b></font></span></p>
373
+ <p style="border: none; padding: 0cm"><br/>
374
+ <br/>
375
+
376
+ </p>
377
+ <h1 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
378
+ <span style="display: inline-block; border: none; padding: 0cm">Orbital
379
+ Mechanics: A Comprehensive Guide</span></h1>
380
+ <h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
381
+ <span style="display: inline-block; border: none; padding: 0cm">Table
382
+ of Contents</span></h2>
383
+ <ol>
384
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Fundamentals
385
+ &amp; Assumptions</span></p></li>
386
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Kepler's
387
+ Laws</span></p></li>
388
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
389
+ Elements</span></p></li>
390
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Vis-Viva
391
+ Equation</span></p></li>
392
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-Analytical
393
+ Methods</span></p></li>
394
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Numerical
395
+ Integration</span></p></li>
396
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Perturbations
397
+ &amp; Stability</span></p></li>
398
+ </ol>
399
+ <hr/>
400
+
401
+ <h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
402
+ <span style="display: inline-block; border: none; padding: 0cm">1.
403
+ Fundamentals and Assumptions</span></h2>
404
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">1.1
405
+ Two-Body Problem</span></h3>
406
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
407
+ fundamental problem in orbital mechanics: two point masses moving
408
+ under their mutual gravity.</span></p>
409
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Assumptions:</span></h4>
410
+ <ol>
411
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Inverse
412
+ Square Law</strong>&nbsp;- Gravitational force varies inversely with
413
+ square of distance</span></p>
414
+ <ul>
415
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">F
416
+ = G * (m₁m₂) / r²</span></p></li>
417
+ </ul>
418
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Universal
419
+ Gravitational Constant</strong>&nbsp;- Same everywhere in the
420
+ universe</span></p>
421
+ <ul>
422
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">G
423
+ = 6.67408 × 10⁻¹¹ N*m²/kg²</span></p></li>
424
+ </ul>
425
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Negligible
426
+ External Forces</strong>&nbsp;- No other gravitational influences
427
+ present</span></p></li>
428
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Non-Rotating
429
+ Reference Frame</strong>&nbsp;- No additional angular momentum</span></p></li>
430
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>No
431
+ Atmospheric Drag</strong>&nbsp;- Vacuum environment</span></p></li>
432
+ </ol>
433
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">1.2
434
+ Orbital Parameters</span></h3>
435
+ <table cellpadding="2" cellspacing="2">
436
+ <thead>
437
+ <tr>
438
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
439
+ Parameter</p>
440
+ </th>
441
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
442
+ Description</p>
443
+ </th>
444
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
445
+ Range</p>
446
+ </th>
447
+ </tr>
448
+ </thead>
449
+ <tbody>
450
+ <tr>
451
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>a</strong></span></p>
452
+ </td>
453
+ <td style="border: none; padding: 0cm"><p>Semi-major axis</p>
454
+ </td>
455
+ <td style="border: none; padding: 0cm"><p>[0, ∞)</p>
456
+ </td>
457
+ </tr>
458
+ <tr>
459
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>e</strong></span></p>
460
+ </td>
461
+ <td style="border: none; padding: 0cm"><p>Eccentricity</p>
462
+ </td>
463
+ <td style="border: none; padding: 0cm"><p>[0, 1]</p>
464
+ </td>
465
+ </tr>
466
+ <tr>
467
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>T</strong></span></p>
468
+ </td>
469
+ <td style="border: none; padding: 0cm"><p>Orbital period</p>
470
+ </td>
471
+ <td style="border: none; padding: 0cm"><p>[0, ∞)</p>
472
+ </td>
473
+ </tr>
474
+ <tr>
475
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>r</strong></span></p>
476
+ </td>
477
+ <td style="border: none; padding: 0cm"><p>Radial distance</p>
478
+ </td>
479
+ <td style="border: none; padding: 0cm"><p>[0, a]</p>
480
+ </td>
481
+ </tr>
482
+ <tr>
483
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>v</strong></span></p>
484
+ </td>
485
+ <td style="border: none; padding: 0cm"><p>Orbital velocity</p>
486
+ </td>
487
+ <td style="border: none; padding: 0cm"><p>[0, v₀ = √(GM/a)]</p>
488
+ </td>
489
+ </tr>
490
+ </tbody>
491
+ </table>
492
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Notation:</span></h4>
493
+ <ul>
494
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>m</strong>&nbsp;and&nbsp;<strong>M</strong>&nbsp;-
495
+ Masses of orbiting body and central body</span></p></li>
496
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>G</strong>&nbsp;-
497
+ Gravitational constant (6.67408 × 10⁻¹¹ N*m²/kg²)</span></p></li>
498
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>μ</strong>&nbsp;=
499
+ GM - Reduced gravitational parameter</span></p></li>
500
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>r</strong>&nbsp;-
501
+ Radial distance from center to orbiting object</span></p></li>
502
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>a</strong>&nbsp;-
503
+ Semi-major axis of orbit</span></p></li>
504
+ </ul>
505
+ <hr/>
506
+
507
+ <h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
508
+ <span style="display: inline-block; border: none; padding: 0cm">2.
509
+ Kepler's Laws of Planetary Motion</span></h2>
510
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">2.1
511
+ First Law: Law of Ellipses</span></h3>
512
+ <p style="margin-right: 1cm; margin-left: 1cm; border: none; padding: 0cm">
513
+ <span style="display: inline-block; border: none; padding: 0cm">&quot;An
514
+ ellipse is described whose foci are the centers of the two bodies,
515
+ and the line connecting the two bodies passes through one focus and
516
+ sweeps out equal areas in equal times.&quot;</span></p>
517
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Mathematical
518
+ Formulation:</span></h4>
519
+ <ul>
520
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Eccentricity</strong>
521
+ (e): Ratio of distance from focus to vertex</span></p>
522
+ <ul>
523
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">e
524
+ = c/a (where c = ae)</span></p></li>
525
+ </ul>
526
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Focus
527
+ Position</strong>: Focus F₁ is at a distance c from center O</span></p>
528
+ <ul>
529
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Position
530
+ vector: r₁ = c * (cos(θ) î + sin(θ) ĵ)</span></p></li>
531
+ </ul>
532
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Focus
533
+ F₂</strong>: At distance c from center O in opposite direction</span></p>
534
+ <ul>
535
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">r₂
536
+ = -c * (cos(θ) î + sin(θ) ĵ)</span></p></li>
537
+ </ul>
538
+ </ul>
539
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Proof:</span></h4>
540
+ <table cellpadding="2" cellspacing="2">
541
+ <thead>
542
+ <tr>
543
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
544
+ Step</p>
545
+ </th>
546
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
547
+ Description</p>
548
+ </th>
549
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
550
+ Formula</p>
551
+ </th>
552
+ </tr>
553
+ </thead>
554
+ <tbody>
555
+ <tr>
556
+ <td style="border: none; padding: 0cm"><p>1</p>
557
+ </td>
558
+ <td style="border: none; padding: 0cm"><p>Define ellipse with foci
559
+ at O and O'</p>
560
+ </td>
561
+ <td style="border: none; padding: 0cm"><p>x²/a² + y²/b² = 1</p>
562
+ </td>
563
+ </tr>
564
+ </tbody>
565
+ </table>
566
+ <table cellpadding="2" cellspacing="2">
567
+ <thead>
568
+ <tr>
569
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
570
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
571
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
572
+ </tr>
573
+ </thead>
574
+ <tbody>
575
+ <tr>
576
+ <td style="border: none; padding: 0cm"><p>2</p>
577
+ </td>
578
+ <td style="border: none; padding: 0cm"><p>Parameterize with angle
579
+ θ</p>
580
+ </td>
581
+ <td style="border: none; padding: 0cm"><p>x = a * cos(θ), y = b *
582
+ sin(θ)</p>
583
+ </td>
584
+ </tr>
585
+ <tr>
586
+ <td style="border: none; padding: 0cm"><p>3</p>
587
+ </td>
588
+ <td style="border: none; padding: 0cm"><p>Apply trigonometric
589
+ identity</p>
590
+ </td>
591
+ <td style="border: none; padding: 0cm"><p>cos²(θ) + sin²(θ) =
592
+ 1</p>
593
+ </td>
594
+ </tr>
595
+ </tbody>
596
+ </table>
597
+ <table cellpadding="2" cellspacing="2">
598
+ <thead>
599
+ <tr>
600
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
601
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
602
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
603
+ </tr>
604
+ </thead>
605
+ <tbody>
606
+ <tr>
607
+ <td style="border: none; padding: 0cm"><p>4</p>
608
+ </td>
609
+ <td style="border: none; padding: 0cm"><p>Express semi-major axis
610
+ (a) and eccentricity (e)</p>
611
+ </td>
612
+ <td style="border: none; padding: 0cm"><p>a² = b²/a² → b² =
613
+ a²(1-e²)</p>
614
+ </td>
615
+ </tr>
616
+ </tbody>
617
+ </table>
618
+ <table cellpadding="2" cellspacing="2">
619
+ <thead>
620
+ <tr>
621
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
622
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
623
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
624
+ </tr>
625
+ </thead>
626
+ <tbody>
627
+ <tr>
628
+ <td style="border: none; padding: 0cm"><p>5</p>
629
+ </td>
630
+ <td style="border: none; padding: 0cm"><p>Derive equation of
631
+ ellipse in standard form</p>
632
+ </td>
633
+ <td style="border: none; padding: 0cm"><p>x²/a² + y²/b² = 1</p>
634
+ </td>
635
+ </tr>
636
+ </tbody>
637
+ </table>
638
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
639
+ Equations:</span></h4>
640
+ <table cellpadding="2" cellspacing="2">
641
+ <thead>
642
+ <tr>
643
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
644
+ Equation</p>
645
+ </th>
646
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
647
+ Description</p>
648
+ </th>
649
+ </tr>
650
+ </thead>
651
+ <tbody>
652
+ <tr>
653
+ <td style="border: none; padding: 0cm"><p>r = a * (1 - e²)/[1 + e
654
+ * cos(θ)]</p>
655
+ </td>
656
+ <td style="border: none; padding: 0cm"><p>Radial distance from
657
+ focus</p>
658
+ </td>
659
+ </tr>
660
+ <tr>
661
+ <td style="border: none; padding: 0cm"><p>T = 2π√(a³/μ)</p>
662
+ </td>
663
+ <td style="border: none; padding: 0cm"><p>Orbital period for
664
+ elliptical orbit</p>
665
+ </td>
666
+ </tr>
667
+ </tbody>
668
+ </table>
669
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">2.2
670
+ Second Law: Equal Area Rule</span></h3>
671
+ <p style="margin-right: 1cm; margin-left: 1cm; border: none; padding: 0cm">
672
+ <span style="display: inline-block; border: none; padding: 0cm">&quot;The
673
+ line connecting the planet to the Sun sweeps out equal areas in equal
674
+ times.&quot;</span></p>
675
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Mathematical
676
+ Proof:</span></h4>
677
+ <ol>
678
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Angular
679
+ Momentum</strong>&nbsp;Conservation: L = r × m v = constant</span></p></li>
680
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Area
681
+ Sweep</strong>: dA = (1/2)r × v dθ = (1/2)r² dθ/v</span></p></li>
682
+ </ol>
683
+ <table cellpadding="2" cellspacing="2">
684
+ <thead>
685
+ <tr>
686
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
687
+ Step</p>
688
+ </th>
689
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
690
+ Description</p>
691
+ </th>
692
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
693
+ Formula</p>
694
+ </th>
695
+ </tr>
696
+ </thead>
697
+ <tbody>
698
+ <tr>
699
+ <td style="border: none; padding: 0cm"><p>3</p>
700
+ </td>
701
+ <td style="border: none; padding: 0cm"><p>Relate angular momentum
702
+ to area sweep</p>
703
+ </td>
704
+ <td style="border: none; padding: 0cm"><p>dA/dt = r²/(2v) dθ/dt</p>
705
+ </td>
706
+ </tr>
707
+ </tbody>
708
+ </table>
709
+ <table cellpadding="2" cellspacing="2">
710
+ <thead>
711
+ <tr>
712
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
713
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
714
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
715
+ </tr>
716
+ </thead>
717
+ <tbody>
718
+ <tr>
719
+ <td style="border: none; padding: 0cm"><p>4</p>
720
+ </td>
721
+ <td style="border: none; padding: 0cm"><p>Substitute v from energy
722
+ equation</p>
723
+ </td>
724
+ <td style="border: none; padding: 0cm"><p>dA/dt = r²/(2v) *
725
+ (GM/r²)/r</p>
726
+ </td>
727
+ </tr>
728
+ </tbody>
729
+ </table>
730
+ <table cellpadding="2" cellspacing="2">
731
+ <thead>
732
+ <tr>
733
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
734
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
735
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
736
+ </tr>
737
+ </thead>
738
+ <tbody>
739
+ <tr>
740
+ <td style="border: none; padding: 0cm"><p>5</p>
741
+ </td>
742
+ <td style="border: none; padding: 0cm"><p>Simplify to show
743
+ proportional relationship</p>
744
+ </td>
745
+ <td style="border: none; padding: 0cm"><p>dA/dt ∝ r/v ∝ 1/r</p>
746
+ </td>
747
+ </tr>
748
+ </tbody>
749
+ </table>
750
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Geometric
751
+ Interpretation:</span></h4>
752
+ <ul>
753
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Area
754
+ swept out is proportional to time elapsed</span></p></li>
755
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Rate
756
+ of area sweep decreases with increasing radius</span></p></li>
757
+ </ul>
758
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">2.3
759
+ Third Law: Harmonies of the Spheres</span></h3>
760
+ <p style="margin-right: 1cm; margin-left: 1cm; border: none; padding: 0cm">
761
+ <span style="display: inline-block; border: none; padding: 0cm">&quot;The
762
+ square of the orbital period of a planet is proportional to the cube
763
+ of its semi-major axis.&quot;</span></p>
764
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Mathematical
765
+ Formulation:</span></h4>
766
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">T²
767
+ = (4π²/GM) * a³</span></p>
768
+ <table cellpadding="2" cellspacing="2">
769
+ <thead>
770
+ <tr>
771
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
772
+ Planet</p>
773
+ </th>
774
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
775
+ Semi-major Axis (AU)</p>
776
+ </th>
777
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
778
+ Orbital Period (Years)</p>
779
+ </th>
780
+ </tr>
781
+ </thead>
782
+ <tbody>
783
+ <tr>
784
+ <td style="border: none; padding: 0cm"><p>Mercury</p>
785
+ </td>
786
+ <td style="border: none; padding: 0cm"><p>0.387</p>
787
+ </td>
788
+ <td style="border: none; padding: 0cm"><p>0.241</p>
789
+ </td>
790
+ </tr>
791
+ <tr>
792
+ <td style="border: none; padding: 0cm"><p>Venus</p>
793
+ </td>
794
+ <td style="border: none; padding: 0cm"><p>0.723</p>
795
+ </td>
796
+ <td style="border: none; padding: 0cm"><p>0.615</p>
797
+ </td>
798
+ </tr>
799
+ <tr>
800
+ <td style="border: none; padding: 0cm"><p>Earth</p>
801
+ </td>
802
+ <td style="border: none; padding: 0cm"><p>1.000</p>
803
+ </td>
804
+ <td style="border: none; padding: 0cm"><p>1.000</p>
805
+ </td>
806
+ </tr>
807
+ </tbody>
808
+ </table>
809
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Universal
810
+ Gravitational Constant (G):</span></h4>
811
+ <ul>
812
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">G
813
+ = 39.478 × 10⁻¹³ km³/Sec²/gm² ≈ 6.67408 × 10⁻¹¹ m³
814
+ kg⁻¹ s⁻²</span></p></li>
815
+ </ul>
816
+ <h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Derived
817
+ Values:</span></h4>
818
+ <ul>
819
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Solar
820
+ mass: Mₛ = 1.989 × 10³⁰ kg</span></p></li>
821
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
822
+ speed: v = √(GM/r) ≈ 29.78 km/s at Earth's orbit</span></p></li>
823
+ </ul>
824
+ <hr/>
825
+
826
+ <h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
827
+ <span style="display: inline-block; border: none; padding: 0cm">3.
828
+ Orbital Elements</span></h2>
829
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.1
830
+ Semi-Analytical Method Overview</span></h3>
831
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-analytical
832
+ methods provide a powerful approach for solving two-body orbital
833
+ problems. These techniques combine numerical and analytical methods
834
+ to accurately compute orbital trajectories.</span></p>
835
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
836
+ semi-analytical method involves three key steps:</span></p>
837
+ <ol>
838
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Numerical
839
+ Integration</strong>: Discretize the orbit into smaller time
840
+ intervals and apply numerical integration techniques to approximate
841
+ the position of the object at each step.</span></p></li>
842
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Interpolation</strong>:
843
+ Use interpolation methods to estimate the position of the object
844
+ between discrete time points.</span></p></li>
845
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Analytical
846
+ Correction</strong>: Apply analytical corrections to refine the
847
+ position and velocity of the object based on Kepler's equations.</span></p></li>
848
+ </ol>
849
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
850
+ numerical integration techniques include Runge-Kutta methods, Verlet
851
+ algorithms, and symplectic integrators. These approaches enable
852
+ detailed simulations of celestial body interactions and gravitational
853
+ influences.</span></p>
854
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.2
855
+ Semi-Analytical Formulation</span></h3>
856
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
857
+ semi-analytical method uses a combination of numerical and analytical
858
+ techniques to compute orbital trajectories. The approach involves
859
+ three primary steps:</span></p>
860
+ <ol>
861
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Numerical
862
+ Integration</strong>: Discretize the orbit into smaller time
863
+ intervals and apply numerical integration techniques such as
864
+ Runge-Kutta or Verlet methods.</span></p></li>
865
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Analytical
866
+ Correction</strong>: Use Kepler's equations to refine the position
867
+ and velocity of the object at each discrete time point.</span></p></li>
868
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Interpolation</strong>:
869
+ Apply interpolation techniques such as cubic spline or Lagrange
870
+ interpolation to estimate the position of the object between
871
+ discrete time points.</span></p></li>
872
+ </ol>
873
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
874
+ semi-analytical method provides a robust approach for computing
875
+ orbital trajectories with high accuracy and efficiency. By combining
876
+ numerical and analytical techniques, this approach can capture
877
+ complex orbital dynamics while minimizing computational resources.</span></p>
878
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.3
879
+ Key Semi-Analytical Formulas</span></h3>
880
+ <table cellpadding="2" cellspacing="2">
881
+ <thead>
882
+ <tr>
883
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
884
+ Formula</p>
885
+ </th>
886
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
887
+ Description</p>
888
+ </th>
889
+ </tr>
890
+ </thead>
891
+ <tbody>
892
+ <tr>
893
+ <td style="border: none; padding: 0cm"><p>r = a * (1 - e²)/(1 + e
894
+ * cos(θ))</p>
895
+ </td>
896
+ <td style="border: none; padding: 0cm"><p>Radial distance from
897
+ focus</p>
898
+ </td>
899
+ </tr>
900
+ <tr>
901
+ <td style="border: none; padding: 0cm"><p>v = √(GM/a) * (1 -
902
+ e²)/(1 + e * cos(θ))</p>
903
+ </td>
904
+ <td style="border: none; padding: 0cm"><p>Orbital velocity</p>
905
+ </td>
906
+ </tr>
907
+ <tr>
908
+ <td style="border: none; padding: 0cm"><p>T = 2π√(a³/μ)</p>
909
+ </td>
910
+ <td style="border: none; padding: 0cm"><p>Orbital period</p>
911
+ </td>
912
+ </tr>
913
+ <tr>
914
+ <td style="border: none; padding: 0cm"><p>f = arctan((r² + b² -
915
+ a²)/2ab)</p>
916
+ </td>
917
+ <td style="border: none; padding: 0cm"><p>True anomaly</p>
918
+ </td>
919
+ </tr>
920
+ <tr>
921
+ <td style="border: none; padding: 0cm"><p>E = √(GM/a) * (1 -
922
+ e²)/(1 + e * cos(θ))</p>
923
+ </td>
924
+ <td style="border: none; padding: 0cm"><p>Eccentric anomaly</p>
925
+ </td>
926
+ </tr>
927
+ </tbody>
928
+ </table>
929
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">These
930
+ formulas provide essential relationships between orbital elements and
931
+ enable accurate computation of celestial body motion.</span></p>
932
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.4
933
+ Semi-Analytical Implementation</span></h3>
934
+ <pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
935
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import odeint</code></span>
936
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
937
+
938
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
939
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
940
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
941
+
942
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">def vis_viva(y, t):</code></span>
943
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">&quot;&quot;&quot;</code></span>
944
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">Vis-viva equation: v² = GM/r</code></span>
945
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">&quot;&quot;&quot;</code></span>
946
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">r = y[0] # Radial distance</code></span>
947
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">v = np.sqrt(G * M_sun / r)</code></span>
948
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
949
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">return [v]</code></span>
950
+
951
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Initial conditions</code></span>
952
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">r0 = 1.496e11 # Average distance from Sun (meters)</code></span>
953
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">v0 = 29.78e3 # Orbital velocity (m/s)</code></span>
954
+
955
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Time array</code></span>
956
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">t = np.linspace(0, 10 * 365.25 * 24 * 3600, 10000) # 10 years</code></span>
957
+
958
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Initial state vector</code></span>
959
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">y0 = [r0]</code></span>
960
+
961
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Solve ODE</code></span>
962
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">solution = odeint(vis_viva, y0, t)</code></span>
963
+
964
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot radial distance over time</code></span>
965
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
966
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/3600/24/365.25, solution[:, 0]/1e9)</code></span>
967
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
968
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Radial Distance (Billion Meters)')</code></span>
969
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Motion')</code></span>
970
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
971
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
972
+ <span style="display: inline-block; border: none; padding: 0cm">This
973
+ Python code implements the semi-analytical method using numerical
974
+ integration and analytical corrections. The </span><span style="display: inline-block; border: 1px solid #3b3b3b; padding: 0.05cm"><code class="western">vis_viva</code></span><span style="display: inline-block; border: none; padding: 0cm">
975
+ function computes the orbital velocity based on Kepler's equation,
976
+ demonstrating a fundamental relationship in celestial mechanics.</span></p>
977
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
978
+ code then generates a 10-year simulation of Earth's orbital motion
979
+ around the Sun. By analyzing the radial distance over time, we can
980
+ visualize the complex elliptical shape of Earth's orbit.</span></p>
981
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.5
982
+ Implementation Details</span></h3>
983
+ <table cellpadding="2" cellspacing="2">
984
+ <thead>
985
+ <tr>
986
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
987
+ Parameter</p>
988
+ </th>
989
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
990
+ Description</p>
991
+ </th>
992
+ <th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
993
+ Range</p>
994
+ </th>
995
+ </tr>
996
+ </thead>
997
+ <tbody>
998
+ <tr>
999
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>a</strong></span></p>
1000
+ </td>
1001
+ <td style="border: none; padding: 0cm"><p>Semi-major axis</p>
1002
+ </td>
1003
+ <td style="border: none; padding: 0cm"><p>[0, ∞)</p>
1004
+ </td>
1005
+ </tr>
1006
+ <tr>
1007
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>e</strong></span></p>
1008
+ </td>
1009
+ <td style="border: none; padding: 0cm"><p>Eccentricity</p>
1010
+ </td>
1011
+ <td style="border: none; padding: 0cm"><p>[0, 1]</p>
1012
+ </td>
1013
+ </tr>
1014
+ <tr>
1015
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>T</strong></span></p>
1016
+ </td>
1017
+ <td style="border: none; padding: 0cm"><p>Orbital period</p>
1018
+ </td>
1019
+ <td style="border: none; padding: 0cm"><p>[0, ∞)</p>
1020
+ </td>
1021
+ </tr>
1022
+ <tr>
1023
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>r</strong></span></p>
1024
+ </td>
1025
+ <td style="border: none; padding: 0cm"><p>Radial distance</p>
1026
+ </td>
1027
+ <td style="border: none; padding: 0cm"><p>[0, a]</p>
1028
+ </td>
1029
+ </tr>
1030
+ <tr>
1031
+ <td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>v</strong></span></p>
1032
+ </td>
1033
+ <td style="border: none; padding: 0cm"><p>Orbital velocity</p>
1034
+ </td>
1035
+ <td style="border: none; padding: 0cm"><p>[0, v₀ = √(GM/a)]</p>
1036
+ </td>
1037
+ </tr>
1038
+ </tbody>
1039
+ </table>
1040
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">These
1041
+ orbital parameters define the fundamental characteristics of an
1042
+ elliptical orbit. By understanding these elements, we can predict the
1043
+ precise motion of celestial bodies and model complex astronomical
1044
+ phenomena.</span></p>
1045
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
1046
+ semi-analytical method offers a powerful approach for analyzing
1047
+ orbital mechanics. By combining numerical integration with analytical
1048
+ corrections, researchers can accurately simulate complex celestial
1049
+ dynamics while minimizing computational resources.</span></p>
1050
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.6
1051
+ Key Implementation Steps</span></h3>
1052
+ <ol>
1053
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Define
1054
+ Orbital Parameters</strong>: Specify the semi-major axis,
1055
+ eccentricity, and initial conditions for the orbit.</span></p></li>
1056
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Discretize
1057
+ Time Array</strong>: Divide the simulation into smaller time
1058
+ intervals to enable numerical integration.</span></p></li>
1059
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Implement
1060
+ Numerical Integration</strong>: Use techniques like Runge-Kutta or
1061
+ Verlet algorithms to approximate the position of the object at each
1062
+ time step.</span></p></li>
1063
+ <li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Apply
1064
+ Analytical Corrections</strong>: Refine the position and velocity
1065
+ using Kepler's equations to achieve high accuracy.</span></p></li>
1066
+ <li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Visualize
1067
+ Orbital Trajectory</strong>: Plot the radial distance over time to
1068
+ visualize the complex elliptical shape of the orbit.</span></p></li>
1069
+ </ol>
1070
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">By
1071
+ following these implementation steps, researchers can leverage the
1072
+ semi-analytical method to model and analyze intricate celestial
1073
+ mechanics with precision and efficiency.</span></p>
1074
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.7
1075
+ Numerical Implementation</span></h3>
1076
+ <pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
1077
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import odeint</code></span>
1078
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
1079
+
1080
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
1081
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
1082
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
1083
+
1084
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">def orbital_equations(state, t):</code></span>
1085
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">&quot;&quot;&quot;</code></span>
1086
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">Orbital equations: dX/dt = v, dV/dt = -GM/r³</code></span>
1087
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">&quot;&quot;&quot;</code></span>
1088
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
1089
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">X, Y, Z, Vx, Vy, Vz = state</code></span>
1090
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
1091
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">r = np.sqrt(X**2 + Y**2 + Z**2)</code></span>
1092
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
1093
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western"># Compute acceleration components</code></span>
1094
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">ax = -G * M_sun * X / r**3</code></span>
1095
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">ay = -G * M_sun * Y / r**3</code></span>
1096
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">az = -G * M_sun * Z / r**3</code></span>
1097
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
1098
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">return [Vx, Vy, Vz, ax, ay, az]</code></span>
1099
+
1100
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Initial conditions</code></span>
1101
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">X0, Y0, Z0 = 1.496e11, 0, 0 # Position in meters</code></span>
1102
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">Vx0, Vy0, Vz0 = 0, 29.78e3, 0 # Velocity in m/s</code></span>
1103
+
1104
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">state0 = [X0, Y0, Z0, Vx0, Vy0, Vz0]</code></span>
1105
+
1106
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Time array</code></span>
1107
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">t = np.linspace(0, 10 * 365.25 * 24 * 3600, 10000) # 10 years</code></span>
1108
+
1109
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Solve ODE</code></span>
1110
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">solution = odeint(orbital_equations, state0, t)</code></span>
1111
+
1112
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Extract position components</code></span>
1113
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">X = solution[:, 0]</code></span>
1114
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">Y = solution[:, 1]</code></span>
1115
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">Z = solution[:, 2]</code></span>
1116
+
1117
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot 3D trajectory</code></span>
1118
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">fig = plt.figure(figsize=(10,6))</code></span>
1119
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax = fig.add_subplot(projection='3d')</code></span>
1120
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.plot(X/1e9, Y/1e9, Z/1e9, label='Earth Orbit')</code></span>
1121
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_xlabel('X (Billion Meters)')</code></span>
1122
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_ylabel('Y (Billion Meters)')</code></span>
1123
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_zlabel('Z (Billion Meters)')</code></span>
1124
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_title('Earth\'s Orbital Motion around Sun')</code></span>
1125
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.legend()</code></span>
1126
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
1127
+ <span style="display: inline-block; border: none; padding: 0cm">This
1128
+ code implements numerical integration using the </span><span style="display: inline-block; border: 1px solid #3b3b3b; padding: 0.05cm"><code class="western">odeint</code></span><span style="display: inline-block; border: none; padding: 0cm">
1129
+ function from SciPy. By solving the system of differential equations
1130
+ that govern orbital motion, we can accurately simulate Earth's 3D
1131
+ trajectory over a 10-year period.</span></p>
1132
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
1133
+ resulting 3D plot visualizes the complex elliptical shape of Earth's
1134
+ orbit around the Sun. By analyzing the position components in x, y,
1135
+ and z directions, we can gain insights into celestial mechanics and
1136
+ orbital dynamics.</span></p>
1137
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.8
1138
+ Analytical Approach</span></h3>
1139
+ <pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
1140
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import quad</code></span>
1141
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
1142
+
1143
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
1144
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
1145
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
1146
+
1147
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">def orbital_energy(r):</code></span>
1148
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">&quot;&quot;&quot;</code></span>
1149
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">Orbital energy: E = -GM/r</code></span>
1150
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">&quot;&quot;&quot;</code></span>
1151
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">return -G * M_sun / r</code></span>
1152
+
1153
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Radial distance array in meters</code></span>
1154
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">r = np.linspace(1.0e11, 1.5e11, 1000)</code></span>
1155
+
1156
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Compute energy at each radial point</code></span>
1157
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">energy = orbital_energy(r)</code></span>
1158
+
1159
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot potential energy curve</code></span>
1160
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
1161
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(r/1e9, energy, label='Potential Energy')</code></span>
1162
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Radial Distance (Billion Meters)')</code></span>
1163
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Energy (Joules)')</code></span>
1164
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Orbital Potential Energy Curve')</code></span>
1165
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1166
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1167
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
1168
+ <span style="display: inline-block; border: none; padding: 0cm">This
1169
+ analytical approach computes the potential energy at each radial
1170
+ distance using Kepler's equation. By visualizing the potential energy
1171
+ curve, we can understand how celestial bodies orbit around massive
1172
+ centers of gravity.</span></p>
1173
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
1174
+ plot reveals a characteristic parabolic shape that represents the
1175
+ gravitational potential energy between two objects. This fundamental
1176
+ relationship underlies orbital mechanics and enables precise
1177
+ predictions of celestial motion.</span></p>
1178
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.9
1179
+ Visualization Techniques</span></h3>
1180
+ <pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
1181
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import odeint</code></span>
1182
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
1183
+
1184
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
1185
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
1186
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
1187
+
1188
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Orbital parameters</code></span>
1189
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">a = 1.496e11 # Average distance from Sun (meters)</code></span>
1190
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">e = 0.0167 # Eccentricity</code></span>
1191
+
1192
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Semi-major axis</code></span>
1193
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">b = a * np.sqrt(1 - e**2)</code></span>
1194
+
1195
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">def orbital_equations(state, t):</code></span>
1196
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">&quot;&quot;&quot;</code></span>
1197
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">Orbital equations: dX/dt = vx, dVx/dt = -GM/r³</code></span>
1198
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">&quot;&quot;&quot;</code></span>
1199
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
1200
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">X, Y, Z, Vx, Vy, Vz = state</code></span>
1201
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
1202
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">r = np.sqrt(X**2 + Y**2 + Z**2)</code></span>
1203
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">v = np.sqrt(G * M_sun / r)</code></span>
1204
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
1205
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western"># Compute acceleration components</code></span>
1206
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">ax = -G * M_sun * X / r**3</code></span>
1207
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">ay = -G * M_sun * Y / r**3</code></span>
1208
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">az = -G * M_sun * Z / r**3</code></span>
1209
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
1210
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">return [Vx, Vy, Vz, ax, ay, az]</code></span>
1211
+
1212
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Initial conditions</code></span>
1213
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">state0 = [a, 0, 0, 0, 29.78e3, 0]</code></span>
1214
+
1215
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Time array</code></span>
1216
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">t = np.linspace(0, 10 * 365.25 * 24 * 3600, 10000) # 10 years</code></span>
1217
+
1218
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Solve ODE</code></span>
1219
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">solution = odeint(orbital_equations, state0, t)</code></span>
1220
+
1221
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot trajectory</code></span>
1222
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">X = solution[:, 0]</code></span>
1223
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">Y = solution[:, 1]</code></span>
1224
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">Z = solution[:, 2]</code></span>
1225
+
1226
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
1227
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(X/1e9, Y/1e9, label='Earth Orbit')</code></span>
1228
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('X (Billion Meters)')</code></span>
1229
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Y (Billion Meters)')</code></span>
1230
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Motion around Sun')</code></span>
1231
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1232
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1233
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
1234
+
1235
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot 3D trajectory</code></span>
1236
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">fig = plt.figure(figsize=(10,6))</code></span>
1237
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax = fig.add_subplot(projection='3d')</code></span>
1238
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.plot(X/1e9, Y/1e9, Z/1e9, label='Orbit')</code></span>
1239
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_xlabel('X (Billion Meters)')</code></span>
1240
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_ylabel('Y (Billion Meters)')</code></span>
1241
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_zlabel('Z (Billion Meters)')</code></span>
1242
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_title('Earth\'s Orbital Motion in 3D')</code></span>
1243
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.legend()</code></span>
1244
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
1245
+
1246
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot radial distance</code></span>
1247
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
1248
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, X/1e9, label='Radial Distance')</code></span>
1249
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
1250
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance from Sun (Billion Meters)')</code></span>
1251
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Radius')</code></span>
1252
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1253
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1254
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
1255
+
1256
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot orbital speed</code></span>
1257
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
1258
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">v = np.sqrt(G * M_sun / X)</code></span>
1259
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, v/1e3, label='Orbital Speed')</code></span>
1260
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
1261
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Speed (km/s)')</code></span>
1262
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Velocity')</code></span>
1263
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1264
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1265
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
1266
+
1267
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot orbital energy</code></span>
1268
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
1269
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">E = -G * M_sun / X</code></span>
1270
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, E/1e30, label='Orbital Energy')</code></span>
1271
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
1272
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Energy (Joules)')</code></span>
1273
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Potential Energy')</code></span>
1274
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1275
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1276
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
1277
+
1278
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot orbital semi-major axis</code></span>
1279
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
1280
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, X/1e9, label='Semi-major Axis')</code></span>
1281
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
1282
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance from Sun (Billion Meters)')</code></span>
1283
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Semi-major Axis')</code></span>
1284
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1285
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1286
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
1287
+
1288
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot orbital eccentricity</code></span>
1289
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
1290
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, e*np.ones(len(t)), label='Eccentricity')</code></span>
1291
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
1292
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Eccentricity')</code></span>
1293
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Eccentricity (Constant)')</code></span>
1294
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1295
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1296
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
1297
+ <span style="display: inline-block; border: none; padding: 0cm">This
1298
+ code generates a comprehensive set of visualizations that capture
1299
+ Earth's orbital dynamics. Each plot provides unique insights into the
1300
+ planet's motion, revealing patterns and characteristics of its
1301
+ celestial behavior.</span></p>
1302
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
1303
+ 3D trajectory visualization is particularly striking, demonstrating
1304
+ Earth's elliptical orbit around the Sun with remarkable detail. By
1305
+ analyzing these plots, researchers can deepen their understanding of
1306
+ orbital mechanics and celestial motion.</span></p>
1307
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
1308
+ radial distance plot shows how Earth's distance from the Sun varies
1309
+ over time, while the orbital speed graph illustrates how velocity
1310
+ changes throughout the orbit. The energy plot reveals the potential
1311
+ energy distribution across Earth's trajectory.</span></p>
1312
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">These
1313
+ visualizations collectively provide a powerful tool for exploring
1314
+ complex astronomical phenomena, enabling scientists to better
1315
+ comprehend the intricate dance of celestial bodies in our solar
1316
+ system.</span></p>
1317
+ <h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.10
1318
+ Orbital Parameter Visualization</span></h3>
1319
+ <pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
1320
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import odeint</code></span>
1321
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
1322
+
1323
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
1324
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
1325
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
1326
+
1327
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Semi-major axis array in meters</code></span>
1328
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">a = np.linspace(1.0e11, 1.5e11, 1000)</code></span>
1329
+
1330
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Compute orbital parameters</code></span>
1331
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">e = 0.0167 * np.ones(len(a))</code></span>
1332
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">T = 365.25 * 24 * 3600 * np.sqrt(a**3 / (G * M_sun))</code></span>
1333
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">r_min = a * (1 - e)</code></span>
1334
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">r_max = a * (1 + e)</code></span>
1335
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">v_min = np.sqrt(G * M_sun / r_max)</code></span>
1336
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">v_max = np.sqrt(G * M_sun / r_min)</code></span>
1337
+
1338
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Create plot</code></span>
1339
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(12,10))</code></span>
1340
+
1341
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Semi-major axis plot</code></span>
1342
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 1)</code></span>
1343
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(a/1e9, label='Semi-major Axis')</code></span>
1344
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Index')</code></span>
1345
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance (Billion Meters)')</code></span>
1346
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Orbital Semi-major Axis')</code></span>
1347
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1348
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1349
+
1350
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Eccentricity plot</code></span>
1351
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 2)</code></span>
1352
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(e*np.ones(len(a)), label='Eccentricity')</code></span>
1353
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Index')</code></span>
1354
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Eccentricity')</code></span>
1355
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Orbital Eccentricity (Constant)')</code></span>
1356
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1357
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1358
+
1359
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Period plot</code></span>
1360
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 3)</code></span>
1361
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(T/365.25, label='Orbital Period')</code></span>
1362
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Semi-major Axis (Billion Meters)')</code></span>
1363
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Period (Years)')</code></span>
1364
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Kepler\'s Third Law')</code></span>
1365
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1366
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1367
+
1368
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Minimum radius plot</code></span>
1369
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 4)</code></span>
1370
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(r_min/1e9, label='Minimum Radius')</code></span>
1371
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Index')</code></span>
1372
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance (Billion Meters)')</code></span>
1373
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Perihelion Distance')</code></span>
1374
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1375
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1376
+
1377
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Maximum radius plot</code></span>
1378
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 5)</code></span>
1379
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(r_max/1e9, label='Maximum Radius')</code></span>
1380
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Index')</code></span>
1381
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance (Billion Meters)')</code></span>
1382
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Aphelion Distance')</code></span>
1383
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1384
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1385
+
1386
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Velocity range plot</code></span>
1387
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 6)</code></span>
1388
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(v_min, v_max, label='Orbital Velocity Range')</code></span>
1389
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Minimum Velocity')</code></span>
1390
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Maximum Velocity')</code></span>
1391
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Orbital Speed Range')</code></span>
1392
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
1393
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
1394
+
1395
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.tight_layout()</code></span>
1396
+ <span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
1397
+ <span style="display: inline-block; border: none; padding: 0cm">This
1398
+ visualization provides a comprehensive overview of orbital
1399
+ parameters. The semi-major axis plot shows how distance from the Sun
1400
+ varies across different orbits, while the eccentricity plot remains
1401
+ constant indicating perfect circles.</span></p>
1402
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
1403
+ period plot demonstrates Kepler's Third Law, revealing an inverse
1404
+ square relationship between orbital period and semi-major axis.
1405
+ Minimum and maximum radius plots represent perihelion and aphelion
1406
+ distances, respectively.</span></p>
1407
+ <p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
1408
+ velocity range plot illustrates how orbital speeds vary between
1409
+ minimum and maximum velocities. By analyzing these visualizations,
1410
+ researchers can gain insights into the fundamental dynamics of
1411
+ celestial motion.</span></p>
1412
+ <p style="line-height: 100%; margin-bottom: 0cm"><br/>
chat_template.jinja ADDED
@@ -0,0 +1,105 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {{- bos_token }}
2
+ {%- if custom_tools is defined %}
3
+ {%- set tools = custom_tools %}
4
+ {%- endif %}
5
+ {%- if not tools_in_user_message is defined %}
6
+ {%- set tools_in_user_message = true %}
7
+ {%- endif %}
8
+ {%- if not date_string is defined %}
9
+ {%- set date_string = "30 Dec 2025" %}
10
+ {%- endif %}
11
+ {%- if not tools is defined %}
12
+ {%- set tools = none %}
13
+ {%- endif %}
14
+ {#- This block extracts the system message, so we can slot it into the right place. #}
15
+ {%- if messages[0]['role'] == 'system' %}
16
+ {%- set system_message = messages[0]['content']|trim %}
17
+ {%- set messages = messages[1:] %}
18
+ {%- else %}
19
+ {%- set system_message = "" %}
20
+ {%- endif %}
21
+ {#- System message + builtin tools #}
22
+ {{- "<|start_header_id|>system<|end_header_id|>\n\n" }}
23
+ {%- if builtin_tools is defined or tools is not none %}
24
+ {{- "Environment: ipython\n" }}
25
+ {%- endif %}
26
+ {%- if builtin_tools is defined %}
27
+ {{- "Tools: " + builtin_tools | reject('equalto', 'code_interpreter') | join(", ") + "\n\n"}}
28
+ {%- endif %}
29
+ {{- "Cutting Knowledge Date: December 2023\n" }}
30
+ {{- "Today Date: " + date_string + "\n\n" }}
31
+ {%- if tools is not none and not tools_in_user_message %}
32
+ {{- "You have access to the following functions. To call a function, please respond with JSON for a function call." }}
33
+ {{- 'Respond in the format {"name": function name, "parameters": dictionary of argument name and its value}.' }}
34
+ {{- "Do not use variables.\n\n" }}
35
+ {%- for t in tools %}
36
+ {{- t | tojson(indent=4) }}
37
+ {{- "\n\n" }}
38
+ {%- endfor %}
39
+ {%- endif %}
40
+ {{- system_message }}
41
+ {{- "<|eot_id|>" }}
42
+ {#- Custom tools are passed in a user message with some extra guidance #}
43
+ {%- if tools_in_user_message and not tools is none %}
44
+ {#- Extract the first user message so we can plug it in here #}
45
+ {%- if messages | length != 0 %}
46
+ {%- set first_user_message = messages[0]['content']|trim %}
47
+ {%- set messages = messages[1:] %}
48
+ {%- else %}
49
+ {{- raise_exception("Cannot put tools in the first user message when there's no first user message!") }}
50
+ {%- endif %}
51
+ {{- '<|start_header_id|>user<|end_header_id|>\n\n' -}}
52
+ {{- "Given the following functions, please respond with a JSON for a function call " }}
53
+ {{- "with its proper arguments that best answers the given prompt.\n\n" }}
54
+ {{- 'Respond in the format {"name": function name, "parameters": dictionary of argument name and its value}.' }}
55
+ {{- "Do not use variables.\n\n" }}
56
+ {%- for t in tools %}
57
+ {{- t | tojson(indent=4) }}
58
+ {{- "\n\n" }}
59
+ {%- endfor %}
60
+ {{- first_user_message + "<|eot_id|>"}}
61
+ {%- endif %}
62
+ {%- for message in messages %}
63
+ {%- if not (message.role == 'ipython' or message.role == 'tool' or 'tool_calls' in message) %}
64
+ {{- '<|start_header_id|>' + message['role'] + '<|end_header_id|>\n\n'+ message['content'] | trim + '<|eot_id|>' }}
65
+ {%- elif 'tool_calls' in message %}
66
+ {%- if not message.tool_calls|length == 1 %}
67
+ {{- raise_exception("This model only supports single tool-calls at once!") }}
68
+ {%- endif %}
69
+ {%- set tool_call = message.tool_calls[0].function %}
70
+ {%- if builtin_tools is defined and tool_call.name in builtin_tools %}
71
+ {{- '<|start_header_id|>assistant<|end_header_id|>\n\n' -}}
72
+ {{- "<|python_tag|>" + tool_call.name + ".call(" }}
73
+ {%- for arg_name, arg_val in tool_call.arguments | items %}
74
+ {{- arg_name + '="' + arg_val + '"' }}
75
+ {%- if not loop.last %}
76
+ {{- ", " }}
77
+ {%- endif %}
78
+ {%- endfor %}
79
+ {{- ")" }}
80
+ {%- else %}
81
+ {{- '<|start_header_id|>assistant<|end_header_id|>\n\n' -}}
82
+ {{- '{"name": "' + tool_call.name + '", ' }}
83
+ {{- '"parameters": ' }}
84
+ {{- tool_call.arguments | tojson }}
85
+ {{- "}" }}
86
+ {%- endif %}
87
+ {%- if builtin_tools is defined %}
88
+ {#- This means we're in ipython mode #}
89
+ {{- "<|eom_id|>" }}
90
+ {%- else %}
91
+ {{- "<|eot_id|>" }}
92
+ {%- endif %}
93
+ {%- elif message.role == "tool" or message.role == "ipython" %}
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1996
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1997
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1998
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1999
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2000
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2001
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2002
+ },
2003
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2004
+ "content": "<|reserved_special_token_245|>",
2005
+ "lstrip": false,
2006
+ "normalized": false,
2007
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2008
+ "single_word": false,
2009
+ "special": true
2010
+ },
2011
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2012
+ "content": "<|reserved_special_token_246|>",
2013
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2014
+ "normalized": false,
2015
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2016
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2017
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2018
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2019
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2020
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2021
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2022
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2023
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2024
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2025
+ "special": true
2026
+ },
2027
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2028
+ "content": "<|reserved_special_token_248|>",
2029
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2030
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2031
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2032
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2033
+ "special": true
2034
+ },
2035
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2036
+ "content": "<|reserved_special_token_249|>",
2037
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2038
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2039
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2040
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2041
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2042
+ },
2043
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2044
+ "content": "<|reserved_special_token_250|>",
2045
+ "lstrip": false,
2046
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2047
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2048
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2049
+ "special": true
2050
+ }
2051
+ },
2052
+ "bos_token": "<|begin_of_text|>",
2053
+ "clean_up_tokenization_spaces": true,
2054
+ "eos_token": "<|eot_id|>",
2055
+ "extra_special_tokens": {},
2056
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2057
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2058
+ "attention_mask"
2059
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2060
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2061
+ "tokenizer_class": "PreTrainedTokenizerFast"
2062
+ }