Gravitation and the two-body problem

Almost every trajectory flown to date is designed on top of one equation, and corrected afterwards for everything that equation leaves out. This page derives that equation, states the quantities it conserves, and quantifies what it ignores.

Newton’s law

Two point masses attract along the line joining them with a force proportional to each mass and inversely proportional to the square of their separation:

where $r$ is the separation, $\hat{\mathbf{r}}$ points from body 1 to body 2, and $G$ is the gravitational constant.

$G$ is the least precisely known of the fundamental constants, measured to about 22 parts per million. That imprecision never propagates into astrodynamics, because no calculation uses $G$ alone.

Why the gravitational parameter, not GM

The observable quantity is not $G$ and not $M$, but their product. A spacecraft orbiting Earth responds to $GM_\oplus$; nothing in its motion separates the two factors. So the product is measured directly, by tracking, and it is known far more precisely than either factor:

G

6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻² (relative uncertainty 2.2 × 10⁻⁵)

μ for Earth

398 600.4418 km³/s² (relative uncertainty ~10⁻⁹)

μ for the Sun

1.327 124 400 18 × 10¹¹ km³/s²

μ for the Moon

4902.800 km³/s²

μ for Mars

42 828.37 km³/s²

Writing $\mu = GM$ is not shorthand. It is a statement about what was measured. Substituting $G$ and a mass estimate in place of a published $\mu$ throws away four orders of magnitude of accuracy.

Reducing the two-body problem

Take two bodies of mass $m_1$ and $m_2$ at inertial positions $\mathbf{R}_1$ and $\mathbf{R}_2$. Newton’s second law gives two coupled equations:

with $\mathbf{r} = \mathbf{R}_2 - \mathbf{R}_1$. Dividing each by its mass and subtracting collapses six degrees of freedom into three:

Two consequences follow immediately, and both are used constantly.

First, the strict $\mu$ in this equation is $G(m_1 + m_2)$, not $GM$ of the primary. For a spacecraft around Earth the spacecraft mass is negligible and the distinction vanishes. For the Earth-Moon system it does not: the correct parameter is $\mu_\oplus + \mu_{\text{Moon}} = 403\,503.2$ km³/s², and using $\mu_\oplus$ alone puts the sidereal month off by about half a day.

Second, the barycentre moves at constant velocity, because the internal forces cancel. The two-body problem therefore has no bearing on where the system as a whole goes.

The constants of the motion

Three vector or scalar quantities are conserved, and between them they contain the entire solution.

Specific angular momentum

Cross $\mathbf{r}$ into the equation of motion. The right side vanishes, because $\mathbf{r} \times \mathbf{r} = 0$, leaving $\mathrm{d}(\mathbf{r} \times \dot{\mathbf{r}})/\mathrm{d}t = 0$, so

$\mathbf{h}$ is normal to the plane containing $\mathbf{r}$ and $\mathbf{v}$. Since it is constant in direction as well as magnitude, the motion is planar. This is why an orbit needs only two angles to orient it, and why a plane change is a separate and expensive manoeuvre: nothing in the two-body problem will produce one for free.

Specific energy

Dot $\dot{\mathbf{r}}$ into the equation of motion and integrate:

The sign of $\varepsilon$ classifies the orbit: negative is bound (ellipse or circle), zero is parabolic, positive is hyperbolic. See Escape velocity and characteristic energy for what follows from that.

The eccentricity vector

This one is peculiar to the inverse-square law. Its constancy is why orbits close on themselves at all: under any other power law the apsidal line precesses. It points from the focus toward periapsis, and its magnitude is the eccentricity.

The orbit equation

Dotting $\mathbf{r}$ into the eccentricity vector and rearranging gives the trajectory in polar form:

with $\nu$ the true anomaly measured from periapsis and $p = a(1-e^2)$ the semi-latus rectum. This is the polar equation of a conic section with the primary at a focus, which is Kepler’s first law recovered from dynamics rather than from data.

Combining the energy integral with $\varepsilon = -\mu/2a$ gives the single most-used relation in the subject:

Speed anywhere on an orbit depends only on where you are and on the semi-major axis. Eccentricity does not appear. Every impulsive manoeuvre calculation in Orbital Mechanics reduces to applying this twice and differencing.

Where one body ends and the next begins

Real trajectories cross between gravitational regimes. Two radii mark the transition, and they answer different questions.

The Hill sphere is the region within which a satellite of the secondary can remain bound against the primary’s tidal pull. It is the stability radius:

The sphere of influence is smaller, and is a modelling boundary rather than a physical one. It is the radius at which the perturbation of the primary on a secondary-centred orbit equals the perturbation of the secondary on a primary-centred orbit:

Patched-conic mission design chains two-body solutions together at the sphere of influence: heliocentric outside, planetocentric inside, with the state vector rotated and re-referenced at the boundary. The discontinuity this introduces is an artefact of the method, not of the physics, and is removed later by numerical integration of the full force model.

What the model leaves out

The two-body equation is the leading term of a much larger sum. The table below gives the order of magnitude of each remaining term for a 400 km circular Earth orbit, where the central acceleration is 8.68 m/s².

Two observations are worth drawing out.

$J_2$ is not a small correction in any operational sense. It is three orders of magnitude below the central term, but it acts secularly: it precesses the node and the apsidal line continuously rather than averaging out. Over weeks it dominates the difference between predicted and actual position. Sun-synchronous orbits exist precisely because this precession can be tuned. See Orbital perturbations.

Drag is the only term in the list that removes energy. Everything else redistributes it. That asymmetry is why orbits below roughly 600 km have finite lifetimes and why the post-mission disposal rules discussed in Space debris and traffic management are written in terms of altitude.

References

• Bate, R. R., Mueller, D. D., White, J. E. and Saylor, W. W. Fundamentals of Astrodynamics, 2nd ed., Dover, 2020.
• Vallado, D. A. Fundamentals of Astrodynamics and Applications, 5th ed., Microcosm Press, 2022.
• Battin, R. H. An Introduction to the Mathematics and Methods of Astrodynamics, revised ed., AIAA, 1999.
• Park, R. S., Folkner, W. M., Williams, J. G. and Boggs, D. H. “The JPL Planetary and Lunar Ephemerides DE440 and DE441”, The Astronomical Journal, 161:105, 2021. Source of record for the $\mu$ values quoted above.