3.1 Two-Body Problem

Fundamental orbital motion under ideal gravitational two-body assumptions

3.1 Two-Body Problem

The two-body problem describes the motion of two masses interacting only through gravitational attraction. In most orbital applications one body, such as a planet or the Sun, is much more massive than the other. The heavier body is therefore treated as the central gravitational source while the smaller body moves under its influence.

This model provides the fundamental analytical framework of orbital mechanics. Under the two-body assumption, the equations of motion can be solved exactly, leading to trajectories that are conic sections. These solutions describe the ideal motion of planets, spacecraft, and satellites before additional perturbations such as atmospheric drag, solar radiation pressure, or third-body gravity are included.

Even in complex mission analysis, the two-body solution remains the reference model used for orbit design, trajectory propagation, and spacecraft navigation.

3.1.1 Two-Body Assumptions

The classical two-body formulation relies on several simplifying physical assumptions.

Newton’s Law of Gravitation

The gravitational force between two masses is governed by Newton’s inverse-square law:

$$F = G \frac{m_1 m_2}{r^2}$$

where $G$ is the gravitational constant and $r$ is the separation between the bodies. In orbital mechanics this is usually expressed using the gravitational parameter

$$\mu = G(m_1 + m_2)$$

For many spacecraft problems, $m_2 \ll m_1$, so

$$\mu \approx Gm_1$$

Point-Mass or Spherical Body Approximation

Each body is treated as a point mass or as a perfectly spherical mass distribution. Outside a spherical body, the gravitational field behaves as if all mass were concentrated at its center. This allows the gravitational force to depend only on the radial distance $r$.

Central Inverse-Square Force

The gravitational force always points toward the central body:

$$\mathbf{F} \parallel -\mathbf{r}$$

The corresponding acceleration becomes

$$\ddot{\mathbf r} = -\frac{\mu}{r^3}\mathbf r$$

Because the force is central:

  • the torque about the center is zero,
  • angular momentum is conserved,
  • motion occurs in a single plane.

Thus all orbital motion can be analyzed within a fixed orbital plane.

3.1.2 Equation of Motion

Applying Newton’s second law to the gravitational force yields the fundamental equation of the two-body problem:

$$\ddot{\mathbf r} = -\frac{\mu}{r^3}\mathbf r$$

where $\mathbf r$ is the position vector of the orbiting body relative to the central mass. This nonlinear differential equation fully describes the orbital dynamics.

Integrals of Motion

Two important quantities remain constant in two-body motion.

Angular Momentum

The specific angular momentum vector is

$$\mathbf h = \mathbf r \times \mathbf v$$

Since no external torque acts on the system,

$$\dot{\mathbf h} = 0$$

so $\mathbf h$ is constant. Its magnitude is

$$h = r^2\dot{\theta}$$

This leads directly to Kepler’s second law: equal areas are swept out in equal times.

Mechanical Energy

The total specific mechanical energy is

$$\varepsilon = \frac{v^2}{2} - \frac{\mu}{r}$$

which remains constant. This constant determines the type of orbit.

Vis-Viva Equation

Rearranging the energy expression gives the fundamental velocity relation

$$v^2 = \mu\left(\frac{2}{r} - \frac{1}{a}\right)$$

where $a$ is the semi-major axis. This equation connects instantaneous velocity with orbital geometry.

3.1.3 Orbital Geometry

The solution to the two-body equation produces trajectories that are conic sections. The general orbit equation is

$$r = \frac{p}{1 + e\cos f}$$

where

  • $p$ : semi-latus rectum
  • $e$ : eccentricity
  • $f$ : true anomaly

Semi-Major Axis and Eccentricity

The semi-major axis $a$ determines orbit size. The eccentricity $e$ determines orbit shape.

$$\mathbf e = \frac{\mathbf v \times \mathbf h}{\mu} - \frac{\mathbf r}{r}$$

Typical classifications are:

Orbit Type Eccentricity
Circular $e = 0$
Elliptical $0 < e < 1$
Parabolic $e = 1$
Hyperbolic $e > 1$

Elliptical orbits correspond to bound motion, while parabolic and hyperbolic trajectories represent escape trajectories.

Periapsis and Apoapsis

The radial distance varies along the orbit:

$$r_p = a(1-e)$$ $$r_a = a(1+e)$$
  • Periapsis: closest approach
  • Apoapsis: farthest distance

Flight-Path Angle

The velocity direction relative to the radial direction is described by the flight-path angle $\gamma$:

$$\tan\gamma = \frac{e\sin f}{1 + e\cos f}$$

This angle determines how much of the velocity is radial versus tangential.

3.1.4 Anomalies and Time

The position of a body along an orbit is described using angular parameters called anomalies.

True Anomaly $f$

The true anomaly measures the angle between the direction of periapsis and the spacecraft’s current position vector. It directly describes the spacecraft’s geometric position along the orbit.

Eccentric Anomaly $E$

For elliptical orbits it is convenient to introduce the eccentric anomaly. Using the auxiliary-circle representation,

$$r = a(1 - e\cos E)$$

This parameter simplifies many orbital equations.

Mean Anomaly and Kepler’s Equation

The mean anomaly

$$M = n(t - t_0)$$

increases linearly with time. The mean motion is

$$n = \sqrt{\frac{\mu}{a^3}}$$

The relation between mean anomaly and eccentric anomaly is Kepler’s equation

$$M = E - e\sin E$$

Because this equation is transcendental, $E$ is usually obtained numerically, for example using Newton iteration.

3.1.5 Position and Velocity Solutions

Once the orbital elements and anomaly are known, the complete spacecraft state can be determined.

Orbital Radius

The radial distance is given by the conic equation

$$r = \frac{p}{1 + e\cos f}$$

where

$$p = \frac{h^2}{\mu}$$

Velocity Components

The velocity vector can be expressed in radial and transverse components.

Radial velocity

$$v_r = \dot r$$

Transverse velocity

$$v_\theta = r\dot\theta$$

Using orbital parameters

$$v_r = \frac{\mu}{h} e\sin f$$ $$v_\theta = \frac{\mu}{h}(1 + e\cos f)$$

State Vector

The position and velocity vectors can be written in the orbital frame as

$$\mathbf r = r(\cos f\,\hat{\mathbf i}_p + \sin f\,\hat{\mathbf j}_p)$$ $$\mathbf v = \dot r\,\hat{\mathbf i}_p + \frac{h}{r}\,\hat{\mathbf j}_p$$

These expressions define the spacecraft state vector $(\mathbf r,\mathbf v)$, which completely describes the orbit at a given time.

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