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_unit: s


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_unit: m/s


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_unit: s


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_unit: s


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_unit: m/s


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_unit: s


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_unit: s


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_unit: s


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_unit: m/s


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_unit: s


topic: astrophysics

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_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 = 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_unit: m/s


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: s


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_unit: J/kg


topic: astrophysics

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_unit: m


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_unit: m


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: m/s


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_unit: m/s


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: m/s


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_unit: s


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: m/s


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_unit: J/kg


topic: astrophysics

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_unit: m


topic: astrophysics

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_unit: m


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_unit: J/kg


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_unit: s


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_unit: s


topic: astrophysics

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_unit: m


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_unit: s


topic: astrophysics

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_unit: m


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_unit: s


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_unit: m


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_unit: m/s


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_unit: s


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_unit: s


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_unit: s


topic: astrophysics

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_unit: W


topic: astrophysics

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_unit: m


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_unit: m


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_unit: s


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: m/s


topic: astrophysics

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_unit: m


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: m/s


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_unit: m/s


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: s


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: s


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_unit: m/s


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_unit: s


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: m


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: J/kg


topic: astrophysics

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_unit: m


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_unit: m


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: s


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_unit: s


topic: astrophysics

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_unit: m


topic: astrophysics

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_unit: m


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_unit: m


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_unit: m


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_unit: m/s


topic: astrophysics

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_unit: m


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: m/s


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_unit: J/kg


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: m/s


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_unit: m


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_unit: s


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_unit: s


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_unit: s


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_unit: m/s


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_unit: m/s


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_unit: m/s


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_unit: m/s


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_unit: J/kg


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_unit: m/s


topic: astrophysics

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_unit: m


topic: astrophysics

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_unit: m


topic: astrophysics

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_unit: m


topic: astrophysics

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_unit: m


topic: astrophysics

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_unit: m


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_unit: s


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_unit: m/s


topic: astrophysics

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_unit: m


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_unit: s


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_unit: m/s


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_unit: s


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_unit: m/s


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_unit: s