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轨道力学·航天
12
engines
Vis-viva · orbital speed
Vis-viva · orbital speed
v²=μ(2/r−1/a) · 任意点速度 · 圆/椭/逃逸+能量
v² = μ(2/r − 1/a) — speed anywhere on a Keplerian orbit
Orbital period · Kepler's third law
Orbital period · Kepler's third law
T=2π√(a³/μ) · 开普勒三定律 · T²∝a³
T = 2π√(a³/μ) — the period grows with the 3/2 power of the semi-major axis
Keplerian orbital elements
Keplerian orbital elements
六根数a/e/i/Ω/ω/ν · 形状+定向 · 近远地点
The six numbers a, e, i, Ω, ω, ν that fix an orbit in space
Kepler's equation · M = E − e·sinE
Kepler's equation · M = E − e·sinE
M=E−e·sinE · 牛顿迭代 · 真近点角+非匀速
Mean → eccentric → true anomaly by Newton iteration
Geostationary orbit · Kepler's third law
Geostationary orbit · Kepler's third law
r=(μT²/4π²)^⅓≈42164km · 高35786km · v≈3.07km/s
Period fixes radius: r = (μT²/4π²)^(1/3), μ = 398600 km³/s²
Lagrange points · restricted three-body problem
Lagrange points · restricted three-body problem
限制性三体L1-L5 · 共线+等边 · L4/L5稳定
Five equilibria in the co-rotating frame of two primaries
霍曼转移 · 两次点火变轨
Hohmann transfer · two-burn orbit change
两次点火 · 转移椭圆 · Δv1/Δv2+转移时间
Minimum-Δv transfer between coplanar circular orbits via a transfer ellipse
双椭圆转移 · 三次点火变轨
Bi-elliptic transfer · three-burn transfer
三次点火 · 中间远地点 · r2/r1≳11.94优于霍曼
Route through a high apoapsis r_b — cheaper than Hohmann when r₂/r₁ is large
轨道倾角改变 · 纯平面变轨 Δv
Inclination change · pure plane change Δv
Δv=2v·sin(Δi/2) · 远地点低速处更省
Rotating the velocity by Δi costs Δv = 2·v·sin(Δi/2) — do it where v is smallest
Tsiolkovsky rocket equation
Tsiolkovsky rocket equation
Δv=Isp·g0·ln(m0/mf) · 质量比 · 分级
Δv = Isp·g0·ln(m0/mf) — the exponential price of mass
Mission Δv budget
Mission Δv budget
任务各段Δv求和 · 火箭方程算质量比
Sum the segments; Tsiolkovsky turns the total into a mass ratio
Gravity assist (flyby)
Gravity assist (flyby)
行星系仅转向 · 日心系最多增2·v行星 · 前/后掠
The planet turns v∞; v_helio = v_planet + v∞ changes the speed