topic: astrophysics

prompt: A star has a measured parallax of p = 0.230 arcsec. Compute its distance in parsecs (d = 1/p). Give your final answer as \boxed{value unit}.

answer: 4.348 pc

gold_unit: m


prompt: For an Earth orbit with semi-major axis a = 18,500 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: -10.77 MJ/kg

gold_unit: J/kg


prompt: A satellite is in an Earth orbit with semi-major axis a = 29,700 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: 14.15 hr

gold_unit: s


prompt: A satellite orbits at altitude h = 1,760 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: 121.6 min

gold_unit: s


prompt: A satellite is in a circular Earth orbit of radius r = 34,100 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: 3.419 km/s

gold_unit: m/s


prompt: An Earth satellite has an orbital period of T = 14.5 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: 2.382e+04 km

gold_unit: m


prompt: A satellite orbits at altitude h = 1,910 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: 125 min

gold_unit: s


prompt: An Earth satellite has an orbital period of T = 5.2 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: 8867 km

gold_unit: m


prompt: Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 10,800 km and r₂ = 33,400 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: 2433 m/s

gold_unit: m/s


prompt: A satellite orbits at altitude h = 1,480 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: 115.4 min

gold_unit: s


prompt: A satellite is in a circular Earth orbit of radius r = 17,800 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: 4.732 km/s

gold_unit: m/s


prompt: A satellite is in an Earth orbit with semi-major axis a = 23,200 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: 9.769 hr

gold_unit: s


prompt: From a distance r = 16,300 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: 6.993 km/s

gold_unit: m/s


topic: astrophysics

prompt: A star has apparent magnitude m = 13.5 and absolute magnitude M = 1.5. 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: 2512 pc

gold_unit: m


topic: astrophysics

prompt: A galaxy recedes at v = 5,500 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: 78.57 Mpc

gold_unit: m


prompt: A spacecraft is on an Earth orbit with semi-major axis a = 34,600 km. At an instant its distance from Earth's center is r = 51,600 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: 1.982 km/s

gold_unit: m/s


prompt: Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 11,300 km and r₂ = 30,500 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: 2192 m/s

gold_unit: m/s


prompt: A satellite is in a circular Earth orbit of radius r = 28,600 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: 3.733 km/s

gold_unit: m/s


topic: astrophysics

prompt: A black hole has mass M = 4.85e+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: 1.433e+07 km

gold_unit: m


topic: astrophysics

prompt: A galaxy recedes at v = 11,800 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: 168.6 Mpc

gold_unit: m


prompt: A spacecraft is on an Earth orbit with semi-major axis a = 19,000 km. At an instant its distance from Earth's center is r = 14,000 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: 5.997 km/s

gold_unit: m/s


topic: astrophysics

prompt: A star has a measured parallax of p = 0.671 arcsec. Compute its distance in parsecs (d = 1/p). Give your final answer as \boxed{value unit}.

answer: 1.49 pc

gold_unit: m


prompt: Compute the total Δv for a Hohmann transfer between two coplanar circular Earth orbits of radii r₁ = 11,600 km and r₂ = 28,300 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: 2011 m/s

gold_unit: m/s


prompt: From a distance r = 14,500 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: 7.415 km/s

gold_unit: m/s


prompt: Two planets orbit the Sun with sidereal periods T₁ = 547.5 days and T₂ = 1,971.0 days. Compute their synodic period (1/T_syn = |1/T₁ − 1/T₂|) in days. Give your final answer as \boxed{value unit}.

answer: 758.1 days

gold_unit: s


topic: astrophysics

prompt: A star has surface temperature T = 26,700 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: 108.5 nm

gold_unit: m


prompt: Two planets orbit the Sun with sidereal periods T₁ = 419.7 days and T₂ = 1,186.2 days. Compute their synodic period (1/T_syn = |1/T₁ − 1/T₂|) in days. Give your final answer as \boxed{value unit}.

answer: 649.6 days

gold_unit: s


topic: astrophysics

prompt: A star has surface temperature T = 15,600 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: 185.8 nm

gold_unit: m


prompt: A spacecraft is on an Earth orbit with semi-major axis a = 9,600 km. At an instant its distance from Earth's center is r = 7,700 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: 7.875 km/s

gold_unit: m/s


prompt: An Earth satellite has an orbital period of T = 18.8 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: 2.952e+04 km

gold_unit: m


prompt: A satellite is in an Earth orbit with semi-major axis a = 25,700 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: 11.39 hr

gold_unit: s


topic: astrophysics

prompt: An exoplanet transit has fractional depth ΔF/F = 0.012140. The host star radius is R_★ = 2.3 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: 27.67 R_earth

gold_unit: m


prompt: For an Earth orbit with semi-major axis a = 25,600 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: -7.785 MJ/kg

gold_unit: J/kg


prompt: A satellite is in a circular Earth orbit of radius r = 13,000 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: 5.537 km/s

gold_unit: m/s


prompt: A spacecraft is on an Earth orbit with semi-major axis a = 9,900 km. At an instant its distance from Earth's center is r = 8,850 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: 7.058 km/s

gold_unit: m/s


prompt: A satellite orbits at altitude h = 1,280 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: 111 min

gold_unit: s


prompt: A satellite is in an Earth orbit with semi-major axis a = 41,200 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: 23.12 hr

gold_unit: s


prompt: From a distance r = 15,600 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: 7.149 km/s

gold_unit: m/s


topic: astrophysics

prompt: A black hole has mass M = 83 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: 245.2 km

gold_unit: m


topic: astrophysics

prompt: A star has apparent magnitude m = 14.5 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: 3162 pc

gold_unit: m


prompt: A satellite orbits at altitude h = 1,480 km (R = 6.371×10^6 m, μ = 3.986×10^14 m³/s²). Report the orbital period in HOURS. Give your final answer as \boxed{value unit}.

answer: 1.923 hr

gold_unit: s


prompt: A probe is in a circular orbit of radius r = 10,800 km about Mars (μ_Mars = 4.283×10^13 m³/s²). Compute the circular speed in km/s. Give your final answer as \boxed{value unit}.

answer: 1.991 km/s

gold_unit: m/s


topic: astrophysics

prompt: A black hole of mass M = 15 M_⊙ (M_⊙ = 1.989×10^30 kg, G = 6.674×10^-11 m³ kg⁻¹ s⁻², c = 2.998×10^8 m/s). Compute the Schwarzschild radius r_s = 2GM/c² in METERS. Give your final answer as \boxed{value unit}.

answer: 4.431e+04 m

gold_unit: m


topic: astrophysics

prompt: A nearby galaxy recedes at v = 4,900 km/s (H₀ = 70 km/s/Mpc). Give its distance in PARSECS (not Mpc). Give your final answer as \boxed{value unit}.

answer: 7e+07 pc

gold_unit: m