2. Gravity, Orbits, and Differential Equations

Space question

Why does a planet keep falling toward its star without hitting it?

An orbit is continuous free fall with sideways velocity. Gravity curves the trajectory while inertia carries the planet forward.

2.1 Position, velocity, acceleration

Position is a vector $\mathbf r(t)$. Its derivatives are

$$\mathbf v=\frac{d\mathbf r}{dt}, \qquad \mathbf a=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}.$$

A derivative is a local rate of change. An integral accumulates those changes. Numerical orbit integration repeatedly uses acceleration to update velocity and velocity to update position.

2.2 Newtonian gravity as a vector equation

For a small body orbiting a mass $M$ at the origin:

$$\frac{d^2\mathbf r}{dt^2} =-\frac{GM}{|\mathbf r|^3}\mathbf r.$$

Why $r^3$ below the vector? The vector $\mathbf r$ has magnitude $r$, so $\mathbf r/r^3=\hat{\mathbf r}/r^2$. The minus sign points acceleration inward.

This is a second-order ordinary differential equation (ODE). Initial position and velocity select one solution from many possible trajectories.

2.3 Circular orbit as a solvable case

Circular motion requires inward acceleration $v^2/r$. Equating this with gravity gives

$$\frac{v^2}{r}=\frac{GM}{r^2} \quad\Rightarrow\quad v_{\rm circ}=\sqrt{\frac{GM}{r}}.$$

The orbital period is circumference divided by speed:

$$T=\frac{2\pi r}{v_{\rm circ}} \quad\Rightarrow\quad T^2=\frac{4\pi^2}{GM}r^3.$$

This is Kepler’s third law derived from Newtonian dynamics.

Worked example: Earth’s scale

Using $GM_\odot=1.327\times10^{20}\ {\rm m^3\,s^{-2}}$ and $r=1.496\times10^{11}$ m gives $v\approx29.8$ km/s and a period near one year.

2.4 Energy explains bound and unbound motion

Specific energy (energy per unit mass) is

$$\epsilon=\frac{v^2}{2}-\frac{GM}{r}.$$

• $\epsilon<0$: bound ellipse;
• $\epsilon=0$: escape boundary;
• $\epsilon>0$: unbound trajectory.

Escape speed follows from $\epsilon=0$:

$$v_{\rm esc}=\sqrt{\frac{2GM}{r}}=\sqrt2\,v_{\rm circ}.$$

Angular momentum $\mathbf h=\mathbf r\times\mathbf v$ is also conserved for an isolated two-body problem. These conservation laws are physical insights and powerful tests of numerical code.

2.5 Computers approximate continuous motion

Euler’s method uses the slope at the beginning of a timestep:

$$\mathbf v_{n+1}=\mathbf v_n+\mathbf a_n\Delta t,\qquad \mathbf r_{n+1}=\mathbf r_n+\mathbf v_n\Delta t.$$

It is simple but tends to create or destroy orbital energy. Velocity Verlet updates position using current acceleration, recomputes acceleration, then updates velocity using the average:

$$\mathbf r_{n+1}=\mathbf r_n+\mathbf v_n\Delta t+ \tfrac12\mathbf a_n\Delta t^2,$$ $$\mathbf v_{n+1}=\mathbf v_n+ \tfrac12(\mathbf a_n+\mathbf a_{n+1})\Delta t.$$

For orbital problems, its long-term conservation behaviour is much better.

Lab

Open 01_orbit_lab.ipynb. You will:

• integrate an orbit in astronomical units and years;
• compare Euler and velocity Verlet;
• measure energy drift rather than judging only by appearance;
• vary initial speed to move among ellipse, circle, and escape.

Used in space science

• predicting planetary, comet, and satellite motion;
• planning gravity assists and transfer orbits;
• inferring masses from the motion of stars, gas, planets, and binaries;
• understanding accretion disks and stellar clusters;
• detecting deviations caused by additional bodies or non-Newtonian effects.

Journey to the frontier

The same habit—write equations of motion, integrate, compare with observations—scales toward frontier problems. Galaxy rotation and gravitational lensing probe dark matter. Precise orbits near compact objects test gravity. Long-term multi-planet stability introduces chaos. Merging compact binaries require general relativity and produce gravitational waves rather than purely Newtonian trajectories.

Newtonian gravity is therefore both a successful theory within its domain and the first rung toward questions it cannot answer.

Connections

• Chapter 1’s distance and angle measurements supply $\mathbf r$.
• Chapter 3 uses Doppler-shifted spectra to supply line-of-sight velocity.
• Chapter 4 uses periodicity to infer an orbit that is not spatially resolved.