pool.jsonl

{"task_id": "astro-orb-leo_period-0000", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 1,030 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "105.6 min", "gold_value_si": 6336.461894323604, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 1030}}
{"task_id": "astro-orb-hohmann_transfer-0001", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 9,300 km and r₂ = 22,600 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "2239 m/s", "gold_value_si": 2238.9992266463796, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 9300, "r2_km": 22600}}
{"task_id": "astro-orb-kepler_third_law-0002", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 18,900 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "7.183 hr", "gold_value_si": 25858.535648485115, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 18900}}
{"task_id": "astro-orb-kepler_third_law-0003", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 44,300 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "25.78 hr", "gold_value_si": 92793.34075359785, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 44300}}
{"task_id": "astro-orb-escape_velocity-0004", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 13,600 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "7.656 km/s", "gold_value_si": 7656.21623642015, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 13600}}
{"task_id": "astro-orb-leo_period-0005", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 390 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "92.21 min", "gold_value_si": 5532.579014289351, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 390}}
{"task_id": "astro-orb-synodic_period-0006", "topic": "orbital_mechanics", "subtopic": "synodic_period", "tier": 2, "prompt": "Two planets orbit the Sun with sidereal periods T₁ = 547.5 days and T₂ = 1,843.2 days. Compute their synodic period (1/T_syn = |1/T₁ − 1/T₂|) in days. Give your final answer as \\boxed{value unit}.", "answer": "778.8 days", "gold_value_si": 67291605.63380282, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"T1_d": 547.5, "T2_d": 1843.25}}
{"task_id": "astro-orb-leo_period-0007", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 480 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "94.06 min", "gold_value_si": 5643.417394649844, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 480}}
{"task_id": "astro-orb-escape_velocity-0008", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 12,300 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "8.051 km/s", "gold_value_si": 8050.6526524301935, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 12300}}
{"task_id": "astro-orb-kepler_third_law-0009", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 39,300 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "21.54 hr", "gold_value_si": 77535.40268036372, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 39300}}
{"task_id": "astro-ap-distance_modulus-0010", "topic": "astrophysics", "subtopic": "distance_modulus", "tier": 2, "prompt": "A star has apparent magnitude m = 20.0 and absolute magnitude M = -2.0. Using the distance modulus m − M = 5·log₁₀(d/10 pc), compute the distance d in parsecs. Give your final answer as \\boxed{value unit}.", "answer": "2.512e+05 pc", "gold_value_si": 7.750871648983153e+21, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m": 20.0, "M": -2.0}}
{"task_id": "astro-orb-hohmann_transfer-0011", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 10,800 km and r₂ = 27,100 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "2130 m/s", "gold_value_si": 2129.7530577325983, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 10800, "r2_km": 27100}}
{"task_id": "astro-orb-hohmann_transfer-0012", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 8,600 km and r₂ = 34,200 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "3048 m/s", "gold_value_si": 3048.2174718669976, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 8600, "r2_km": 34200}}
{"task_id": "astro-ap-hubble_law-0013", "topic": "astrophysics", "subtopic": "hubble_law", "tier": 1, "prompt": "A galaxy recedes at v = 26,900 km/s. Using Hubble's law with H₀ = 70 km/s/Mpc, compute its distance in Mpc. Give your final answer as \\boxed{value unit}.", "answer": "384.3 Mpc", "gold_value_si": 1.1857818134621429e+25, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"v_kms": 26900}}
{"task_id": "astro-orb-kepler_third_law-0014", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 29,900 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "14.29 hr", "gold_value_si": 51453.86511212905, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 29900}}
{"task_id": "astro-orb-specific_orbital_energy-0015", "topic": "orbital_mechanics", "subtopic": "specific_orbital_energy", "tier": 1, "prompt": "For an Earth orbit with semi-major axis a = 13,300 km and μ = 3.986×10^14 m³/s², compute the specific orbital energy ε = −μ/(2a) in MJ/kg. Give your final answer as \\boxed{value unit}.", "answer": "-14.98 MJ/kg", "gold_value_si": -14984962.406015038, "gold_unit": "J/kg", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 13300}}
{"task_id": "astro-ap-transit_radius-0016", "topic": "astrophysics", "subtopic": "transit_radius", "tier": 2, "prompt": "An exoplanet transit has fractional depth ΔF/F = 0.018740. The host star radius is R_★ = 1.8 R_⊙ (R_⊙ = 6.957×10^8 m, R_⊕ = 6.371×10^6 m). Using ΔF/F = (R_p/R_★)², compute the planet radius in Earth radii (R_⊕). Give your final answer as \\boxed{value unit}.", "answer": "26.91 R_earth", "gold_value_si": 171427030.29692838, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"depth": 0.01874, "r_star_rsun": 1.8}}
{"task_id": "astro-orb-altitude_from_period-0017", "topic": "orbital_mechanics", "subtopic": "altitude_from_period", "tier": 3, "prompt": "An Earth satellite has an orbital period of T = 9.9 hours. Using μ = 3.986×10^14 m³/s² and Earth radius R = 6.371×10^6 m, solve Kepler's third law for the semi-major axis, then report the altitude above Earth's surface in km. Give your final answer as \\boxed{value unit}.", "answer": "1.704e+04 km", "gold_value_si": 17036281.75473459, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_hr": 9.9}}
{"task_id": "astro-ap-distance_modulus-0018", "topic": "astrophysics", "subtopic": "distance_modulus", "tier": 2, "prompt": "A star has apparent magnitude m = 13.5 and absolute magnitude M = 1.0. Using the distance modulus m − M = 5·log₁₀(d/10 pc), compute the distance d in parsecs. Give your final answer as \\boxed{value unit}.", "answer": "3162 pc", "gold_value_si": 9.757769282459845e+19, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m": 13.5, "M": 1.0}}
{"task_id": "astro-orb-vis_viva-0019", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 9,400 km. At an instant its distance from Earth's center is r = 9,700 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "6.307 km/s", "gold_value_si": 6307.242796274798, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 9400, "r_km": 9700.0}}
{"task_id": "astro-orb-hohmann_transfer-0020", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 7,700 km and r₂ = 31,700 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "3261 m/s", "gold_value_si": 3261.027119290647, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 7700, "r2_km": 31700}}
{"task_id": "astro-orb-escape_velocity-0021", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 17,700 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "6.711 km/s", "gold_value_si": 6711.151020696738, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 17700}}
{"task_id": "astro-orb-vis_viva-0022", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 39,100 km. At an instant its distance from Earth's center is r = 43,850 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "2.826 km/s", "gold_value_si": 2825.9133450240824, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 39100, "r_km": 43850.0}}
{"task_id": "astro-ap-wien_law-0023", "topic": "astrophysics", "subtopic": "wien_law", "tier": 1, "prompt": "A star has surface temperature T = 15,000 K. Using Wien's displacement law λ_peak = b/T with b = 2.898×10^-3 m·K, compute the peak emission wavelength in nm. Give your final answer as \\boxed{value unit}.", "answer": "193.2 nm", "gold_value_si": 1.932e-07, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_K": 15000}}
{"task_id": "astro-orb-vis_viva-0024", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 33,300 km. At an instant its distance from Earth's center is r = 37,150 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "3.08 km/s", "gold_value_si": 3080.4188403388075, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 33300, "r_km": 37150.0}}
{"task_id": "astro-ap-distance_modulus-0025", "topic": "astrophysics", "subtopic": "distance_modulus", "tier": 2, "prompt": "A star has apparent magnitude m = 6.0 and absolute magnitude M = 0.5. Using the distance modulus m − M = 5·log₁₀(d/10 pc), compute the distance d in parsecs. Give your final answer as \\boxed{value unit}.", "answer": "125.9 pc", "gold_value_si": 3.8846379199539185e+18, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m": 6.0, "M": 0.5}}
{"task_id": "astro-orb-circular_velocity-0026", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 27,600 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "3.8 km/s", "gold_value_si": 3800.2669623997795, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 27600}}
{"task_id": "astro-orb-vis_viva-0027", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 37,500 km. At an instant its distance from Earth's center is r = 64,650 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "1.304 km/s", "gold_value_si": 1304.4845013970655, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 37500, "r_km": 64650.0}}
{"task_id": "astro-orb-kepler_third_law-0028", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 27,300 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "12.47 hr", "gold_value_si": 44890.57742434611, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 27300}}
{"task_id": "astro-ap-wien_law-0029", "topic": "astrophysics", "subtopic": "wien_law", "tier": 1, "prompt": "A star has surface temperature T = 18,900 K. Using Wien's displacement law λ_peak = b/T with b = 2.898×10^-3 m·K, compute the peak emission wavelength in nm. Give your final answer as \\boxed{value unit}.", "answer": "153.3 nm", "gold_value_si": 1.5333333333333333e-07, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_K": 18900}}
{"task_id": "astro-orb-escape_velocity-0030", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 8,600 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "9.628 km/s", "gold_value_si": 9627.963150043972, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 8600}}
{"task_id": "astro-orb-circular_velocity-0031", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 21,400 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "4.316 km/s", "gold_value_si": 4315.804470118991, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 21400}}
{"task_id": "astro-ap-transit_radius-0032", "topic": "astrophysics", "subtopic": "transit_radius", "tier": 2, "prompt": "An exoplanet transit has fractional depth ΔF/F = 0.016690. The host star radius is R_★ = 0.5 R_⊙ (R_⊙ = 6.957×10^8 m, R_⊕ = 6.371×10^6 m). Using ΔF/F = (R_p/R_★)², compute the planet radius in Earth radii (R_⊕). Give your final answer as \\boxed{value unit}.", "answer": "7.054 R_earth", "gold_value_si": 44938665.974025086, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"depth": 0.01669, "r_star_rsun": 0.5}}
{"task_id": "astro-orb-vis_viva-0033", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 10,500 km. At an instant its distance from Earth's center is r = 9,950 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "6.493 km/s", "gold_value_si": 6492.97299033121, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 10500, "r_km": 9950.0}}
{"task_id": "astro-orb-circular_velocity-0034", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 16,700 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "4.886 km/s", "gold_value_si": 4885.515681384503, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 16700}}
{"task_id": "astro-orb-specific_orbital_energy-0035", "topic": "orbital_mechanics", "subtopic": "specific_orbital_energy", "tier": 1, "prompt": "For an Earth orbit with semi-major axis a = 43,100 km and μ = 3.986×10^14 m³/s², compute the specific orbital energy ε = −μ/(2a) in MJ/kg. Give your final answer as \\boxed{value unit}.", "answer": "-4.624 MJ/kg", "gold_value_si": -4624129.930394432, "gold_unit": "J/kg", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 43100}}
{"task_id": "astro-ap-parallax_distance-0036", "topic": "astrophysics", "subtopic": "parallax_distance", "tier": 1, "prompt": "A star has a measured parallax of p = 0.715 arcsec. Compute its distance in parsecs (d = 1/p). Give your final answer as \\boxed{value unit}.", "answer": "1.399 pc", "gold_value_si": 4.315632981118882e+16, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"p_arcsec": 0.715}}
{"task_id": "astro-ap-transit_radius-0037", "topic": "astrophysics", "subtopic": "transit_radius", "tier": 2, "prompt": "An exoplanet transit has fractional depth ΔF/F = 0.016250. The host star radius is R_★ = 2.1 R_⊙ (R_⊙ = 6.957×10^8 m, R_⊕ = 6.371×10^6 m). Using ΔF/F = (R_p/R_★)², compute the planet radius in Earth radii (R_⊕). Give your final answer as \\boxed{value unit}.", "answer": "29.23 R_earth", "gold_value_si": 186237863.46934128, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"depth": 0.01625, "r_star_rsun": 2.1}}
{"task_id": "astro-orb-specific_orbital_energy-0038", "topic": "orbital_mechanics", "subtopic": "specific_orbital_energy", "tier": 1, "prompt": "For an Earth orbit with semi-major axis a = 44,800 km and μ = 3.986×10^14 m³/s², compute the specific orbital energy ε = −μ/(2a) in MJ/kg. Give your final answer as \\boxed{value unit}.", "answer": "-4.449 MJ/kg", "gold_value_si": -4448660.714285715, "gold_unit": "J/kg", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 44800}}
{"task_id": "astro-orb-leo_period-0039", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 1,970 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "126.4 min", "gold_value_si": 7581.207838503933, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 1970}}
{"task_id": "astro-orb-leo_period-0040", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 1,690 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "120 min", "gold_value_si": 7202.688002077256, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 1690}}
{"task_id": "astro-ap-parallax_distance-0041", "topic": "astrophysics", "subtopic": "parallax_distance", "tier": 1, "prompt": "A star has a measured parallax of p = 0.669 arcsec. Compute its distance in parsecs (d = 1/p). Give your final answer as \\boxed{value unit}.", "answer": "1.495 pc", "gold_value_si": 4.6123730665171896e+16, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"p_arcsec": 0.669}}
{"task_id": "astro-orb-synodic_period-0042", "topic": "orbital_mechanics", "subtopic": "synodic_period", "tier": 2, "prompt": "Two planets orbit the Sun with sidereal periods T₁ = 511.0 days and T₂ = 1,277.5 days. Compute their synodic period (1/T_syn = |1/T₁ − 1/T₂|) in days. Give your final answer as \\boxed{value unit}.", "answer": "851.7 days", "gold_value_si": 73583999.99999999, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"T1_d": 510.99999999999994, "T2_d": 1277.5}}
{"task_id": "astro-ap-hubble_law-0043", "topic": "astrophysics", "subtopic": "hubble_law", "tier": 1, "prompt": "A galaxy recedes at v = 23,900 km/s. Using Hubble's law with H₀ = 70 km/s/Mpc, compute its distance in Mpc. Give your final answer as \\boxed{value unit}.", "answer": "341.4 Mpc", "gold_value_si": 1.0535384885407142e+25, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"v_kms": 23900}}
{"task_id": "astro-orb-kepler_third_law-0044", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 19,500 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "7.528 hr", "gold_value_si": 27099.616076718576, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 19500}}
{"task_id": "astro-orb-altitude_from_period-0045", "topic": "orbital_mechanics", "subtopic": "altitude_from_period", "tier": 3, "prompt": "An Earth satellite has an orbital period of T = 18 hours. Using μ = 3.986×10^14 m³/s² and Earth radius R = 6.371×10^6 m, solve Kepler's third law for the semi-major axis, then report the altitude above Earth's surface in km. Give your final answer as \\boxed{value unit}.", "answer": "2.85e+04 km", "gold_value_si": 28498243.32472083, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_hr": 18.0}}
{"task_id": "astro-orb-circular_velocity-0046", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 21,900 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "4.266 km/s", "gold_value_si": 4266.252833812025, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 21900}}
{"task_id": "astro-orb-leo_period-0047", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 650 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "97.58 min", "gold_value_si": 5854.767868623821, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 650}}
{"task_id": "astro-orb-kepler_third_law-0048", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 24,100 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "10.34 hr", "gold_value_si": 37233.7505671123, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 24100}}
{"task_id": "astro-orb-synodic_period-0049", "topic": "orbital_mechanics", "subtopic": "synodic_period", "tier": 2, "prompt": "Two planets orbit the Sun with sidereal periods T₁ = 511.0 days and T₂ = 2,007.5 days. Compute their synodic period (1/T_syn = |1/T₁ − 1/T₂|) in days. Give your final answer as \\boxed{value unit}.", "answer": "685.5 days", "gold_value_si": 59226146.34146341, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"T1_d": 510.99999999999994, "T2_d": 2007.5}}
{"task_id": "astro-ap-stefan_boltzmann-0050", "topic": "astrophysics", "subtopic": "stefan_boltzmann", "tier": 3, "prompt": "A star has radius R = 15.7 R_⊙ (R_⊙ = 6.957×10^8 m) and surface temperature T = 6,100 K. With σ = 5.670×10^-8 W m⁻² K⁻⁴, compute its luminosity L = 4πR²σT⁴ in watts. Give your final answer as \\boxed{value unit}.", "answer": "1.177e+29 W", "gold_value_si": 1.176943390948736e+29, "gold_unit": "W", "rel_tol": 0.02, "hand_curated": false, "params": {"r_rsun": 15.7, "T_K": 6100}}
{"task_id": "astro-ap-parallax_distance-0051", "topic": "astrophysics", "subtopic": "parallax_distance", "tier": 1, "prompt": "A star has a measured parallax of p = 0.796 arcsec. Compute its distance in parsecs (d = 1/p). Give your final answer as \\boxed{value unit}.", "answer": "1.256 pc", "gold_value_si": 3.876479373743718e+16, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"p_arcsec": 0.796}}
{"task_id": "astro-orb-altitude_from_period-0052", "topic": "orbital_mechanics", "subtopic": "altitude_from_period", "tier": 3, "prompt": "An Earth satellite has an orbital period of T = 8.7 hours. Using μ = 3.986×10^14 m³/s² and Earth radius R = 6.371×10^6 m, solve Kepler's third law for the semi-major axis, then report the altitude above Earth's surface in km. Give your final answer as \\boxed{value unit}.", "answer": "1.51e+04 km", "gold_value_si": 15104355.153582223, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_hr": 8.7}}
{"task_id": "astro-orb-synodic_period-0053", "topic": "orbital_mechanics", "subtopic": "synodic_period", "tier": 2, "prompt": "Two planets orbit the Sun with sidereal periods T₁ = 511.0 days and T₂ = 1,916.2 days. Compute their synodic period (1/T_syn = |1/T₁ − 1/T₂|) in days. Give your final answer as \\boxed{value unit}.", "answer": "696.8 days", "gold_value_si": 60205090.90909089, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"T1_d": 510.99999999999994, "T2_d": 1916.25}}
{"task_id": "astro-ap-wien_law-0054", "topic": "astrophysics", "subtopic": "wien_law", "tier": 1, "prompt": "A star has surface temperature T = 15,200 K. Using Wien's displacement law λ_peak = b/T with b = 2.898×10^-3 m·K, compute the peak emission wavelength in nm. Give your final answer as \\boxed{value unit}.", "answer": "190.7 nm", "gold_value_si": 1.906578947368421e-07, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_K": 15200}}
{"task_id": "astro-orb-hohmann_transfer-0055", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 8,700 km and r₂ = 27,000 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "2716 m/s", "gold_value_si": 2715.839868374189, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 8700, "r2_km": 27000}}
{"task_id": "astro-orb-vis_viva-0056", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 15,800 km. At an instant its distance from Earth's center is r = 20,800 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "3.619 km/s", "gold_value_si": 3619.2644246666, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 15800, "r_km": 20800.0}}
{"task_id": "astro-ap-schwarzschild_radius-0057", "topic": "astrophysics", "subtopic": "schwarzschild_radius", "tier": 2, "prompt": "A black hole has mass M = 4.53e+06 solar masses (M_⊙ = 1.989×10^30 kg). With G = 6.674×10^-11 m³ kg⁻¹ s⁻² and c = 2.998×10^8 m/s, compute the Schwarzschild radius r_s = 2GM/c² in km. Give your final answer as \\boxed{value unit}.", "answer": "1.338e+07 km", "gold_value_si": 13380918517.615255, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m_solar": 4530000.0}}
{"task_id": "astro-ap-hubble_law-0058", "topic": "astrophysics", "subtopic": "hubble_law", "tier": 1, "prompt": "A galaxy recedes at v = 1,300 km/s. Using Hubble's law with H₀ = 70 km/s/Mpc, compute its distance in Mpc. Give your final answer as \\boxed{value unit}.", "answer": "18.57 Mpc", "gold_value_si": 5.730544079928572e+23, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"v_kms": 1300}}
{"task_id": "astro-orb-circular_velocity-0059", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 24,400 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "4.042 km/s", "gold_value_si": 4041.7898972819567, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 24400}}
{"task_id": "astro-orb-circular_velocity-0060", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 17,000 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "4.842 km/s", "gold_value_si": 4842.216313169974, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 17000}}
{"task_id": "astro-orb-circular_velocity-0061", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 7,300 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "7.389 km/s", "gold_value_si": 7389.366666097129, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 7300}}
{"task_id": "astro-orb-vis_viva-0062", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 26,000 km. At an instant its distance from Earth's center is r = 40,000 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "2.145 km/s", "gold_value_si": 2144.581723607372, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 26000, "r_km": 40000.0}}
{"task_id": "astro-ap-hubble_law-0063", "topic": "astrophysics", "subtopic": "hubble_law", "tier": 1, "prompt": "A galaxy recedes at v = 9,800 km/s. Using Hubble's law with H₀ = 70 km/s/Mpc, compute its distance in Mpc. Give your final answer as \\boxed{value unit}.", "answer": "140 Mpc", "gold_value_si": 4.3199486140999997e+24, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"v_kms": 9800}}
{"task_id": "astro-orb-leo_period-0064", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 1,660 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "119.4 min", "gold_value_si": 7162.51690558725, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 1660}}
{"task_id": "astro-orb-hohmann_transfer-0065", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 11,100 km and r₂ = 24,600 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "1893 m/s", "gold_value_si": 1893.433702184107, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 11100, "r2_km": 24600}}
{"task_id": "astro-ap-hubble_law-0066", "topic": "astrophysics", "subtopic": "hubble_law", "tier": 1, "prompt": "A galaxy recedes at v = 27,400 km/s. Using Hubble's law with H₀ = 70 km/s/Mpc, compute its distance in Mpc. Give your final answer as \\boxed{value unit}.", "answer": "391.4 Mpc", "gold_value_si": 1.2078223676157143e+25, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"v_kms": 27400}}
{"task_id": "astro-orb-hohmann_transfer-0067", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 8,400 km and r₂ = 21,400 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "2442 m/s", "gold_value_si": 2442.251755077507, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 8400, "r2_km": 21400}}
{"task_id": "astro-ap-hubble_law-0068", "topic": "astrophysics", "subtopic": "hubble_law", "tier": 1, "prompt": "A galaxy recedes at v = 6,500 km/s. Using Hubble's law with H₀ = 70 km/s/Mpc, compute its distance in Mpc. Give your final answer as \\boxed{value unit}.", "answer": "92.86 Mpc", "gold_value_si": 2.865272039964286e+24, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"v_kms": 6500}}
{"task_id": "astro-orb-leo_period-0069", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 1,550 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "116.9 min", "gold_value_si": 7015.865281277513, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 1550}}
{"task_id": "astro-orb-hohmann_transfer-0070", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 10,100 km and r₂ = 38,500 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "2769 m/s", "gold_value_si": 2768.5121376781085, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 10100, "r2_km": 38500}}
{"task_id": "astro-orb-kepler_third_law-0071", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 42,800 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "24.48 hr", "gold_value_si": 88120.4841738047, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 42800}}
{"task_id": "astro-ap-hubble_law-0072", "topic": "astrophysics", "subtopic": "hubble_law", "tier": 1, "prompt": "A galaxy recedes at v = 7,200 km/s. Using Hubble's law with H₀ = 70 km/s/Mpc, compute its distance in Mpc. Give your final answer as \\boxed{value unit}.", "answer": "102.9 Mpc", "gold_value_si": 3.1738397981142856e+24, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"v_kms": 7200}}
{"task_id": "astro-orb-vis_viva-0073", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 29,500 km. At an instant its distance from Earth's center is r = 44,850 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "2.065 km/s", "gold_value_si": 2064.6889592527073, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 29500, "r_km": 44850.0}}
{"task_id": "astro-orb-altitude_from_period-0074", "topic": "orbital_mechanics", "subtopic": "altitude_from_period", "tier": 3, "prompt": "An Earth satellite has an orbital period of T = 6.5 hours. Using μ = 3.986×10^14 m³/s² and Earth radius R = 6.371×10^6 m, solve Kepler's third law for the semi-major axis, then report the altitude above Earth's surface in km. Give your final answer as \\boxed{value unit}.", "answer": "1.131e+04 km", "gold_value_si": 11311205.116730478, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_hr": 6.5}}
{"task_id": "astro-ap-distance_modulus-0075", "topic": "astrophysics", "subtopic": "distance_modulus", "tier": 2, "prompt": "A star has apparent magnitude m = 0.5 and absolute magnitude M = -5.5. Using the distance modulus m − M = 5·log₁₀(d/10 pc), compute the distance d in parsecs. Give your final answer as \\boxed{value unit}.", "answer": "158.5 pc", "gold_value_si": 4.890469393049225e+18, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m": 0.5, "M": -5.5}}
{"task_id": "astro-orb-circular_velocity-0076", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 35,800 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "3.337 km/s", "gold_value_si": 3336.776620076703, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 35800}}
{"task_id": "astro-orb-specific_orbital_energy-0077", "topic": "orbital_mechanics", "subtopic": "specific_orbital_energy", "tier": 1, "prompt": "For an Earth orbit with semi-major axis a = 27,800 km and μ = 3.986×10^14 m³/s², compute the specific orbital energy ε = −μ/(2a) in MJ/kg. Give your final answer as \\boxed{value unit}.", "answer": "-7.169 MJ/kg", "gold_value_si": -7169064.748201439, "gold_unit": "J/kg", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 27800}}
{"task_id": "astro-ap-parallax_distance-0078", "topic": "astrophysics", "subtopic": "parallax_distance", "tier": 1, "prompt": "A star has a measured parallax of p = 0.402 arcsec. Compute its distance in parsecs (d = 1/p). Give your final answer as \\boxed{value unit}.", "answer": "2.488 pc", "gold_value_si": 7.675814879353234e+16, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"p_arcsec": 0.402}}
{"task_id": "astro-orb-altitude_from_period-0079", "topic": "orbital_mechanics", "subtopic": "altitude_from_period", "tier": 3, "prompt": "An Earth satellite has an orbital period of T = 5.9 hours. Using μ = 3.986×10^14 m³/s² and Earth radius R = 6.371×10^6 m, solve Kepler's third law for the semi-major axis, then report the altitude above Earth's surface in km. Give your final answer as \\boxed{value unit}.", "answer": "1.021e+04 km", "gold_value_si": 10205602.65616799, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_hr": 5.9}}
{"task_id": "astro-orb-circular_velocity-0080", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 29,400 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "3.682 km/s", "gold_value_si": 3682.094937566344, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 29400}}
{"task_id": "astro-ap-schwarzschild_radius-0081", "topic": "astrophysics", "subtopic": "schwarzschild_radius", "tier": 2, "prompt": "A black hole has mass M = 93 solar masses (M_⊙ = 1.989×10^30 kg). With G = 6.674×10^-11 m³ kg⁻¹ s⁻² and c = 2.998×10^8 m/s, compute the Schwarzschild radius r_s = 2GM/c² in km. Give your final answer as \\boxed{value unit}.", "answer": "274.7 km", "gold_value_si": 274707.5987060085, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m_solar": 93.0}}
{"task_id": "astro-orb-kepler_third_law-0082", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 20,000 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "7.819 hr", "gold_value_si": 28148.562085893667, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 20000}}
{"task_id": "astro-orb-synodic_period-0083", "topic": "orbital_mechanics", "subtopic": "synodic_period", "tier": 2, "prompt": "Two planets orbit the Sun with sidereal periods T₁ = 419.7 days and T₂ = 1,752.0 days. Compute their synodic period (1/T_syn = |1/T₁ − 1/T₂|) in days. Give your final answer as \\boxed{value unit}.", "answer": "552 days", "gold_value_si": 47692799.99999999, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"T1_d": 419.74999999999994, "T2_d": 1752.0}}
{"task_id": "astro-ap-schwarzschild_radius-0084", "topic": "astrophysics", "subtopic": "schwarzschild_radius", "tier": 2, "prompt": "A black hole has mass M = 57 solar masses (M_⊙ = 1.989×10^30 kg). With G = 6.674×10^-11 m³ kg⁻¹ s⁻² and c = 2.998×10^8 m/s, compute the Schwarzschild radius r_s = 2GM/c² in km. Give your final answer as \\boxed{value unit}.", "answer": "168.4 km", "gold_value_si": 168369.17340045684, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m_solar": 57.0}}
{"task_id": "astro-ap-wien_law-0085", "topic": "astrophysics", "subtopic": "wien_law", "tier": 1, "prompt": "A star has surface temperature T = 25,800 K. Using Wien's displacement law λ_peak = b/T with b = 2.898×10^-3 m·K, compute the peak emission wavelength in nm. Give your final answer as \\boxed{value unit}.", "answer": "112.3 nm", "gold_value_si": 1.1232558139534884e-07, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_K": 25800}}
{"task_id": "astro-orb-altitude_from_period-0086", "topic": "orbital_mechanics", "subtopic": "altitude_from_period", "tier": 3, "prompt": "An Earth satellite has an orbital period of T = 8.3 hours. Using μ = 3.986×10^14 m³/s² and Earth radius R = 6.371×10^6 m, solve Kepler's third law for the semi-major axis, then report the altitude above Earth's surface in km. Give your final answer as \\boxed{value unit}.", "answer": "1.444e+04 km", "gold_value_si": 14440956.7471942, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_hr": 8.3}}
{"task_id": "astro-orb-altitude_from_period-0087", "topic": "orbital_mechanics", "subtopic": "altitude_from_period", "tier": 3, "prompt": "An Earth satellite has an orbital period of T = 11 hours. Using μ = 3.986×10^14 m³/s² and Earth radius R = 6.371×10^6 m, solve Kepler's third law for the semi-major axis, then report the altitude above Earth's surface in km. Give your final answer as \\boxed{value unit}.", "answer": "1.874e+04 km", "gold_value_si": 18739535.618521426, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_hr": 11.0}}
{"task_id": "astro-orb-escape_velocity-0088", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 8,500 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "9.684 km/s", "gold_value_si": 9684.432626339947, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 8500}}
{"task_id": "astro-ap-wien_law-0089", "topic": "astrophysics", "subtopic": "wien_law", "tier": 1, "prompt": "A star has surface temperature T = 15,300 K. Using Wien's displacement law λ_peak = b/T with b = 2.898×10^-3 m·K, compute the peak emission wavelength in nm. Give your final answer as \\boxed{value unit}.", "answer": "189.4 nm", "gold_value_si": 1.8941176470588234e-07, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_K": 15300}}
{"task_id": "astro-ap-parallax_distance-0090", "topic": "astrophysics", "subtopic": "parallax_distance", "tier": 1, "prompt": "A star has a measured parallax of p = 0.265 arcsec. Compute its distance in parsecs (d = 1/p). Give your final answer as \\boxed{value unit}.", "answer": "3.774 pc", "gold_value_si": 1.1644066345283018e+17, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"p_arcsec": 0.265}}
{"task_id": "astro-orb-escape_velocity-0091", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 18,700 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "6.529 km/s", "gold_value_si": 6529.243144712927, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 18700}}
{"task_id": "astro-orb-vis_viva-0092", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 14,000 km. At an instant its distance from Earth's center is r = 20,300 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "3.286 km/s", "gold_value_si": 3286.26039582419, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 14000, "r_km": 20300.0}}
{"task_id": "astro-orb-specific_orbital_energy-0093", "topic": "orbital_mechanics", "subtopic": "specific_orbital_energy", "tier": 1, "prompt": "For an Earth orbit with semi-major axis a = 36,200 km and μ = 3.986×10^14 m³/s², compute the specific orbital energy ε = −μ/(2a) in MJ/kg. Give your final answer as \\boxed{value unit}.", "answer": "-5.506 MJ/kg", "gold_value_si": -5505524.861878453, "gold_unit": "J/kg", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 36200}}
{"task_id": "astro-orb-escape_velocity-0094", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 15,700 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "7.126 km/s", "gold_value_si": 7125.803117101557, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 15700}}
{"task_id": "astro-ap-schwarzschild_radius-0095", "topic": "astrophysics", "subtopic": "schwarzschild_radius", "tier": 2, "prompt": "A black hole has mass M = 74 solar masses (M_⊙ = 1.989×10^30 kg). With G = 6.674×10^-11 m³ kg⁻¹ s⁻² and c = 2.998×10^8 m/s, compute the Schwarzschild radius r_s = 2GM/c² in km. Give your final answer as \\boxed{value unit}.", "answer": "218.6 km", "gold_value_si": 218584.54090585627, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m_solar": 74.0}}
{"task_id": "astro-orb-escape_velocity-0096", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 19,000 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "6.477 km/s", "gold_value_si": 6477.491392263065, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 19000}}
{"task_id": "astro-orb-altitude_from_period-0097", "topic": "orbital_mechanics", "subtopic": "altitude_from_period", "tier": 3, "prompt": "An Earth satellite has an orbital period of T = 6 hours. Using μ = 3.986×10^14 m³/s² and Earth radius R = 6.371×10^6 m, solve Kepler's third law for the semi-major axis, then report the altitude above Earth's surface in km. Give your final answer as \\boxed{value unit}.", "answer": "1.039e+04 km", "gold_value_si": 10392383.734007698, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_hr": 6.0}}
{"task_id": "astro-orb-synodic_period-0098", "topic": "orbital_mechanics", "subtopic": "synodic_period", "tier": 2, "prompt": "Two planets orbit the Sun with sidereal periods T₁ = 492.8 days and T₂ = 1,149.8 days. Compute their synodic period (1/T_syn = |1/T₁ − 1/T₂|) in days. Give your final answer as \\boxed{value unit}.", "answer": "862.3 days", "gold_value_si": 74503800.0, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"T1_d": 492.75000000000006, "T2_d": 1149.75}}
{"task_id": "astro-orb-leo_period-0099", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 1,590 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "117.8 min", "gold_value_si": 7069.076101664404, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 1590}}
{"task_id": "astro-orb-kepler_third_law-0100", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 14,700 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "4.927 hr", "gold_value_si": 17737.288228123318, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 14700}}
{"task_id": "astro-orb-hohmann_transfer-0101", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 11,200 km and r₂ = 20,200 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "1491 m/s", "gold_value_si": 1491.3975843734502, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 11200, "r2_km": 20200}}
{"task_id": "astro-orb-hohmann_transfer-0102", "topic": "orbital_mechanics", "subtopic": "hohmann_transfer", "tier": 3, "prompt": "Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 9,300 km and r₂ = 41,800 km. Use μ = 3.986×10^14 m³/s². Sum the two burns and give the total in m/s. Give your final answer as \\boxed{value unit}.", "answer": "3052 m/s", "gold_value_si": 3051.9403635308618, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r1_km": 9300, "r2_km": 41800}}
{"task_id": "astro-orb-circular_velocity-0103", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 24,000 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "4.075 km/s", "gold_value_si": 4075.3322972897968, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 24000}}
{"task_id": "astro-orb-circular_velocity-0104", "topic": "orbital_mechanics", "subtopic": "circular_velocity", "tier": 1, "prompt": "A satellite is in a circular Earth orbit of radius r = 20,800 km. With μ = 3.986×10^14 m³/s², compute the circular orbital speed v = √(μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "4.378 km/s", "gold_value_si": 4377.609112113773, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 20800}}
{"task_id": "astro-orb-specific_orbital_energy-0105", "topic": "orbital_mechanics", "subtopic": "specific_orbital_energy", "tier": 1, "prompt": "For an Earth orbit with semi-major axis a = 38,700 km and μ = 3.986×10^14 m³/s², compute the specific orbital energy ε = −μ/(2a) in MJ/kg. Give your final answer as \\boxed{value unit}.", "answer": "-5.15 MJ/kg", "gold_value_si": -5149870.801033592, "gold_unit": "J/kg", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 38700}}
{"task_id": "astro-orb-escape_velocity-0106", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 16,400 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "6.972 km/s", "gold_value_si": 6972.069714049119, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 16400}}
{"task_id": "astro-ap-distance_modulus-0107", "topic": "astrophysics", "subtopic": "distance_modulus", "tier": 2, "prompt": "A star has apparent magnitude m = 17.5 and absolute magnitude M = -4.0. Using the distance modulus m − M = 5·log₁₀(d/10 pc), compute the distance d in parsecs. Give your final answer as \\boxed{value unit}.", "answer": "1.995e+05 pc", "gold_value_si": 6.156736194511261e+21, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m": 17.5, "M": -4.0}}
{"task_id": "astro-ap-transit_radius-0108", "topic": "astrophysics", "subtopic": "transit_radius", "tier": 2, "prompt": "An exoplanet transit has fractional depth ΔF/F = 0.021390. The host star radius is R_★ = 1.5 R_⊙ (R_⊙ = 6.957×10^8 m, R_⊕ = 6.371×10^6 m). Using ΔF/F = (R_p/R_★)², compute the planet radius in Earth radii (R_⊕). Give your final answer as \\boxed{value unit}.", "answer": "23.96 R_earth", "gold_value_si": 152622532.1748889, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"depth": 0.02139, "r_star_rsun": 1.5}}
{"task_id": "astro-ap-schwarzschild_radius-0109", "topic": "astrophysics", "subtopic": "schwarzschild_radius", "tier": 2, "prompt": "A black hole has mass M = 31 solar masses (M_⊙ = 1.989×10^30 kg). With G = 6.674×10^-11 m³ kg⁻¹ s⁻² and c = 2.998×10^8 m/s, compute the Schwarzschild radius r_s = 2GM/c² in km. Give your final answer as \\boxed{value unit}.", "answer": "91.57 km", "gold_value_si": 91569.19956866952, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m_solar": 31.0}}
{"task_id": "astro-ap-wien_law-0110", "topic": "astrophysics", "subtopic": "wien_law", "tier": 1, "prompt": "A star has surface temperature T = 28,900 K. Using Wien's displacement law λ_peak = b/T with b = 2.898×10^-3 m·K, compute the peak emission wavelength in nm. Give your final answer as \\boxed{value unit}.", "answer": "100.3 nm", "gold_value_si": 1.0027681660899655e-07, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"T_K": 28900}}
{"task_id": "astro-ap-parallax_distance-0111", "topic": "astrophysics", "subtopic": "parallax_distance", "tier": 1, "prompt": "A star has a measured parallax of p = 0.770 arcsec. Compute its distance in parsecs (d = 1/p). Give your final answer as \\boxed{value unit}.", "answer": "1.299 pc", "gold_value_si": 4.007373482467533e+16, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"p_arcsec": 0.77}}
{"task_id": "astro-orb-leo_period-0112", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 1,720 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "120.7 min", "gold_value_si": 7242.9339191195095, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 1720}}
{"task_id": "astro-orb-vis_viva-0113", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 26,600 km. At an instant its distance from Earth's center is r = 24,300 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "4.222 km/s", "gold_value_si": 4221.566291807448, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 26600, "r_km": 24300.0}}
{"task_id": "astro-ap-distance_modulus-0114", "topic": "astrophysics", "subtopic": "distance_modulus", "tier": 2, "prompt": "A star has apparent magnitude m = 10.5 and absolute magnitude M = 0.0. Using the distance modulus m − M = 5·log₁₀(d/10 pc), compute the distance d in parsecs. Give your final answer as \\boxed{value unit}.", "answer": "1259 pc", "gold_value_si": 3.884637919953918e+19, "gold_unit": "m", "rel_tol": 0.02, "hand_curated": false, "params": {"m": 10.5, "M": 0.0}}
{"task_id": "astro-orb-kepler_third_law-0115", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 15,200 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "5.181 hr", "gold_value_si": 18649.904105835427, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 15200}}
{"task_id": "astro-orb-vis_viva-0116", "topic": "orbital_mechanics", "subtopic": "vis_viva", "tier": 2, "prompt": "A spacecraft is on an Earth orbit with semi-major axis a = 20,400 km. At an instant its distance from Earth's center is r = 15,900 km. With μ = 3.986×10^14 m³/s², use the vis-viva equation v = √(μ(2/r − 1/a)) to find its speed in km/s. Give your final answer as \\boxed{value unit}.", "answer": "5.532 km/s", "gold_value_si": 5531.649762376474, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 20400, "r_km": 15900.0}}
{"task_id": "astro-orb-kepler_third_law-0117", "topic": "orbital_mechanics", "subtopic": "kepler_third_law", "tier": 1, "prompt": "A satellite is in an Earth orbit with semi-major axis a = 14,800 km. Earth's standard gravitational parameter is μ = 3.986×10^14 m³/s². Compute the orbital period in hours. Give your final answer as \\boxed{value unit}.", "answer": "4.977 hr", "gold_value_si": 17918.58842781769, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"a_km": 14800}}
{"task_id": "astro-orb-escape_velocity-0118", "topic": "orbital_mechanics", "subtopic": "escape_velocity", "tier": 1, "prompt": "From a distance r = 11,800 km from Earth's center, with μ = 3.986×10^14 m³/s², compute the escape velocity v = √(2μ/r) in km/s. Give your final answer as \\boxed{value unit}.", "answer": "8.219 km/s", "gold_value_si": 8219.44779373276, "gold_unit": "m/s", "rel_tol": 0.02, "hand_curated": false, "params": {"r_km": 11800}}
{"task_id": "astro-orb-leo_period-0119", "topic": "orbital_mechanics", "subtopic": "leo_period", "tier": 2, "prompt": "A satellite orbits at altitude h = 350 km above Earth's surface. With Earth radius R = 6.371×10^6 m and μ = 3.986×10^14 m³/s², compute its orbital period in minutes. Give your final answer as \\boxed{value unit}.", "answer": "91.39 min", "gold_value_si": 5483.553238373392, "gold_unit": "s", "rel_tol": 0.02, "hand_curated": false, "params": {"h_km": 350}}