gravitation
Fields & potential, two-body reduction, effective potential & orbit classification, ellipse geometry, Kepler's laws, escape.
field & potential · two-body reduction · effective potential · orbits · Kepler & escape
Field & potential
Force & field. \(F=Gm_1m_2/r^2\); \(\vv{g}=\vv{F}/m\), \(g=GM/r^2\) directed toward the mass.
Shell theorem. A uniform spherical shell attracts an external mass as if all its mass sat at the centre, and exerts zero force inside — so any spherically symmetric body acts externally as a point mass, and only the mass interior to \(r\) pulls on a body at radius \(r\).
Potential. \(V_g=-GM/r\); field strength \(g=-\dd V_g/\dd r\); potential energy \(U=mV_g=-GMm/r\).
Near surface. \(g\approx\) constant, \(\Delta U=mg\Delta h\).
Superposition. \(\vv{g}=\sum\vv{g}_i\), \(V=\sum V_i\).
Two-body reduction
One-body problem. Two bodies interacting only with each other: the CM moves uniformly, and the relative coordinate \(\vv r=\vv r_1-\vv r_2\) obeys \(\mu\ddot{\vv r}=\vv F(r)\,\uvec r\) with reduced mass \[\mu=\frac{m_1m_2}{m_1+m_2}.\] For \(m\ll M\), \(\mu\approx m\) and the heavy body sits at the focus; comparable masses orbit the common CM.
General central-force properties. \(\vv L\) about the centre is constant, so the motion stays in a plane; \(E\) and \(L\) are the constants of the motion. Equal areas: \(\dfrac{\dd A}{\dd t}=\dfrac{L}{2\mu}\) — Kepler's second law, true for any central force.
Effective potential
Energy equation. With \(L=\mu r^2\dot\theta\) eliminated, the radial motion is one-dimensional: \[E=\tfrac12\mu\dot r^2+U_{\mathrm{eff}}(r),\qquad U_{\mathrm{eff}}=U(r)+\frac{L^2}{2\mu r^2}.\] The \(L^2/2\mu r^2\) term is the centrifugal barrier — it keeps any orbit with \(L\ne0\) from the centre.
Reading the diagram. Turning points where \(U_{\mathrm{eff}}=E\); circular orbit at the minimum of \(U_{\mathrm{eff}}\); \(E\lt0\) bound (two turning points), \(E\ge0\) unbound (one). Small radial disturbance of a circular orbit oscillates at \(\omega_r^2=U_{\mathrm{eff}}''(r_0)/\mu\); for the inverse-square force \(\omega_r\) equals the orbital frequency, which is why the ellipse closes.
Orbits
Orbit equation. For \(U=-GMm/r\) (write \(C=GMm\)): \[r=\frac{r_0}{1-\varepsilon\cos\theta},\qquad r_0=\frac{L^2}{\mu C},\qquad \varepsilon=\sqrt{1+\frac{2EL^2}{\mu C^2}}.\] \(\varepsilon\) fixes the shape, \(r_0\) only the scale.
| Energy | Eccentricity | Orbit |
|---|---|---|
| \(E\gt0\) | \(\varepsilon\gt1\) | hyperbola (Rutherford scattering) |
| \(E=0\) | \(\varepsilon=1\) | parabola |
| \(E\lt0\) | \(0\lt\varepsilon\lt1\) | ellipse, focus at the centre of force |
| \(E=-\mu C^2/2L^2\) | \(\varepsilon=0\) | circle |
Ellipse geometry. \(r_{\min,\max}=r_0/(1\pm\varepsilon)\), \(r_{\max}/r_{\min}=(1+\varepsilon)/(1-\varepsilon)\); semi-major \(a=r_0/(1-\varepsilon^2)\), semi-minor \(b=a\sqrt{1-\varepsilon^2}=L/\sqrt{2\mu|E|}\); \(\varepsilon\) = (centre–focus distance)/\(a\).
Energy sets the size. \(E=-\dfrac{GMm}{2a}\): the major axis depends on energy alone, so orbits of equal \(a\) (whatever their \(\varepsilon\)) have equal \(E\) and equal period. Tangential thrust changes \(E\), hence \(a\) and \(T\) — the basis of satellite manoeuvres.
Circular orbit. \(GMm/r^2=mv^2/r\), so \(v=\sqrt{GM/r}\); \(T=2\pi r/v=2\pi\sqrt{r^3/(GM)}\); \(E=K+U=-GMm/(2r)\), \(K=-E=GMm/(2r)\).
Kepler & escape
Kepler's laws. (1) Ellipses with the sun at one focus — the signature of the inverse-square force. (2) Equal areas in equal times — angular momentum conservation, any central force. (3) Law of periods: \[T^2=\frac{4\pi^2a^3}{G(M+m)}\approx\frac{4\pi^2}{GM}\,a^3,\] from sweeping the ellipse's area \(\pi ab\) at rate \(L/2\mu\). The \(M+m\) shows \(T^2\propto a^3\) is not exact — the "constant" carries the planet's mass (a \(10^{-3}\) effect even for Jupiter).
Escape speed. \(E\ge0\): \(v_e=\sqrt{2GM/r}=\sqrt2\,v_{\mathrm{circ}}\) — independent of launch direction (energy is a scalar); 11.2 km/s from Earth's surface, ignoring rotation and drag.