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天体物理
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Escape velocity · v = √(2GM/r)
Escape velocity · v = √(2GM/r)
v=√(2GM/r) · 环绕速度 v/√2 · 各天体对比
The speed to break free of a body's gravity, versus the orbital speed
Schwarzschild radius · r_s = 2GM/c²
Schwarzschild radius · r_s = 2GM/c²
rs=2GM/c² · 事件视界 · 太阳→2.95km 地球→9mm
The event-horizon size of a black hole of a given mass
Tidal force · Δa = 2GMΔr/d³
Tidal force · Δa = 2GMΔr/d³
差异引力 Δa=2GMΔr/d³ · 潮汐隆起 · 1/d³ 标度
Why the near Moon out-pulls the distant Sun on Earth's oceans
Stellar luminosity · Stefan–Boltzmann
Stellar luminosity · Stefan–Boltzmann
L=4πR²σT⁴ · L/L☉=(R/R☉)²(T/T☉)⁴ · 恒星类型
L = 4πR²σT⁴ → in solar units L/L☉ = (R/R☉)²·(T/T☉)⁴
Distance modulus · magnitude & distance
Distance modulus · magnitude & distance
m−M=5log₁₀d−5 · 视/绝对星等 · 秒差距/光年
m − M = 5·log₁₀(d) − 5 (d in parsecs)
Planetary equilibrium temperature
Planetary equilibrium temperature
T_eq=T★√(R★/2d)(1−A)^¼ · 地球≈255K · 宜居带
T_eq = T★·√(R★ / 2d)·(1 − A)^(1/4)
Telescope resolution · diffraction limit
Telescope resolution · diffraction limit
瑞利判据 θ=1.22λ/D · 角分辨率 · 口径对比
Rayleigh criterion θ = 1.22 λ/D — a bigger aperture and a shorter wavelength resolve finer detail
Exoplanet transit · light-curve depth
Exoplanet transit · light-curve depth
掩星深度 δ=(Rp/Rs)² · 光变曲线 · 木星~1%/地球~84ppm
Transit depth δ = (R_planet / R_star)² — the fraction of starlight the planet blocks
Gravitational lensing · the Einstein radius
Gravitational lensing · the Einstein radius
爱因斯坦半径 θE ∝ √M · 光环/多重像 · 放大
θ_E = √( 4GM/c² · D_ls/(D_l·D_s) ) — a foreground mass bends a background source into a ring
Roche limit — tidal disruption
Roche limit — tidal disruption
d=2.44R_M(ρM/ρm)^⅓ · 潮汐瓦解 · 行星环
Inside the Roche limit self-gravity loses to the tide and the satellite breaks into a ring
Relativistic Doppler shift
Relativistic Doppler shift
z=√((1+β)/(1−β))−1 · 红移/蓝移 · 对比经典 β
How a spectral line reddens or blueshifts when the source moves along the line of sight
Mean-motion orbital resonance
Mean-motion orbital resonance
周期比 p:q · 拉普拉斯 4:2:1 · 会合周期
When two orbital periods lock into a small-integer ratio, conjunctions repeat and tugs pile up