Celestial Mechanics: Key Equations
Reference compilation of fundamental equations in celestial mechanics, from Newton's law of gravitation to orbital dynamics.
## Fundamental Laws
**Newton's Law of Universal Gravitation:**
$$F = G\frac{m_1 m_2}{r^2}$$
where $G = 6.674 \times 10^{-11}$ N·m²/kg²
**Kepler's Laws:**
1. **First Law (Elliptical Orbits):** Planetary orbits are ellipses with the Sun at one focus
- Polar equation: $r = \frac{a(1-e^2)}{1 + e\cos\theta}$
2. **Second Law (Equal Areas):**
$$\frac{dA}{dt} = \frac{L}{2m} = \text{constant}$$
where $L$ is angular momentum
3. **Third Law (Harmonic Law):**
$$T^2 = \frac{4\pi^2}{G(M+m)}a^3 \approx \frac{4\pi^2}{GM}a^3$$
## Two-Body Problem
**Equation of Motion (Reduced Mass):**
$$\ddot{\mathbf{r}} = -\frac{GM}{r^3}\mathbf{r}$$
**Vis-Viva Equation:**
$$v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right)$$
**Specific Orbital Energy:**
$$\varepsilon = \frac{v^2}{2} - \frac{GM}{r} = -\frac{GM}{2a}$$
**Specific Angular Momentum:**
$$h = r v \sin\gamma = \sqrt{GMa(1-e^2)}$$
## Orbital Elements
**Eccentricity:**
$$e = \sqrt{1 + \frac{2\varepsilon h^2}{(GM)^2}}$$
**Semi-major Axis:**
$$a = -\frac{GM}{2\varepsilon}$$
**Orbital Period:**
$$T = 2\pi\sqrt{\frac{a^3}{GM}}$$
## Position in Orbit
**Kepler's Equation (Elliptic):**
$$M = E - e\sin E$$
where $M = n(t-t_0)$ is mean anomaly, $E$ is eccentric anomaly, $n = \sqrt{GM/a^3}$
**True Anomaly from Eccentric Anomaly:**
$$\tan\frac{\theta}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}$$
**Time Since Periapsis (Elliptic):**
$$t - t_p = \sqrt{\frac{a^3}{GM}}(E - e\sin E)$$
## Velocity Components
**Radial Velocity:**
$$v_r = \sqrt{\frac{GM}{a(1-e^2)}}e\sin\theta$$
**Tangential Velocity:**
$$v_\theta = \sqrt{\frac{GM}{a(1-e^2)}}(1 + e\cos\theta)$$
## Special Cases
**Circular Orbit Velocity:**
$$v_c = \sqrt{\frac{GM}{r}}$$
**Escape Velocity:**
$$v_{esc} = \sqrt{\frac{2GM}{r}} = \sqrt{2}\,v_c$$
**Parabolic Orbit ($e=1$):**
$$v = \sqrt{\frac{2GM}{r}}$$
## Orbital Transfers
**Hohmann Transfer $\Delta v$:**
$$\Delta v_1 = \sqrt{\frac{GM}{r_1}}\left(\sqrt{\frac{2r_2}{r_1+r_2}} - 1\right)$$
$$\Delta v_2 = \sqrt{\frac{GM}{r_2}}\left(1 - \sqrt{\frac{2r_1}{r_1+r_2}}\right)$$
**Transfer Time:**
$$t_{transfer} = \pi\sqrt{\frac{(r_1+r_2)^3}{8GM}}$$
## Lagrange Planetary Equations
**Perturbation equations for orbital elements:**
$$\frac{da}{dt} = \frac{2}{na}\frac{\partial R}{\partial M}$$
$$\frac{de}{dt} = \frac{1-e^2}{nae}\frac{\partial R}{\partial M} - \frac{\sqrt{1-e^2}}{na^2e}\frac{\partial R}{\partial\omega}$$
$$\frac{di}{dt} = \frac{1}{na^2\sqrt{1-e^2}\sin i}\left(\cos i\frac{\partial R}{\partial\omega} - \frac{\partial R}{\partial\Omega}\right)$$
where $R$ is the disturbing function.
## Three-Body Problem
**Jacobi Constant (Circular Restricted):**
$$C_J = 2U - v^2$$
$$U = \frac{1-\mu}{r_1} + \frac{\mu}{r_2} + \frac{1}{2}(x^2 + y^2)$$
**Lagrange Points (Collinear, approximate):**
$$L_1: r \approx a\left(1 - \left(\frac{\mu}{3}\right)^{1/3}\right)$$
## Orbital Perturbations
**J₂ Perturbation (Oblateness):**
$$\dot{\Omega} = -\frac{3}{2}nJ_2\left(\frac{R_e}{a}\right)^2\frac{\cos i}{(1-e^2)^2}$$
$$\dot{\omega} = \frac{3}{4}nJ_2\left(\frac{R_e}{a}\right)^2\frac{4-5\sin^2 i}{(1-e^2)^2}$$
where $J_2 \approx 1.08263 \times 10^{-3}$ for Earth.
## Sphere of Influence
**Laplace's Approximation:**
$$r_{SOI} \approx a\left(\frac{m}{M}\right)^{2/5}$$
These equations form the mathematical foundation for analyzing planetary motion, satellite orbits, spacecraft trajectories, and gravitational interactions throughout the solar system.