1
Two-Body Problem

1.1 Introduction

The two-body problem is the fundamental model of orbital mechanics. It describes the motion of two bodies in pure gravitational interaction to the exclusion of all other forces. In the restricted two-body problem, the mass of one body is assumed to be negligible, while the other is considered as point-like. This modeling is appropriate to the motion of artificial satellites around a celestial body.

The two-body problem has an analytical solution called Keplerian motion. The trajectories are conics, satisfying properties of conservation of energy and angular momentum. The conic nature (circle, ellipse, parabola, or hyperbola) depends only on the initial conditions of position and velocity and the motion time law is determined by Kepler’s equation. The Keplerian orbit of a satellite around the attracting body is represented geometrically by its orbital parameters, which are analytically related to the position and velocity.

The first section presents the dynamic model and derives the main prime integrals of the Keplerian motion, namely those of angular momentum, eccentricity vector, and energy. The conic shape is then determined from the initial conditions of position and velocity.

The second section deals with the time law of the motion. Position and time are linked by Kepler’s equation, which is transcendental. Kepler’s equation takes different forms depending on the conic nature. It can also be expressed in a universal form valid for any kind of orbit. The solution of Kepler’s equation requires a numerical iterative method.

The third section defines the orbital parameters locating the orbit in space and establishes their relation to position and velocity. The classical orbital parameters have singularities for circular or equatorial orbits, which motivates the definition of the equinoctial parameters, devoid of such singularities. An overview of Earth’s orbits is finally presented, with a focus on Earth’s coordinate systems and on specific properties such as geo-synchronism or sun-synchronism.

1.2 Keplerian Motion

The two-body problem, or Kepler’s problem, concerns the relative motion of two particles exerting between them the force of gravitational attraction. The resulting dynamics, called Keplerian motion, is presented in this section.

1.2.1 Dynamic Model

The two-body problem is formulated in a Galilean (or inertial) reference frame.

Both bodies are considered as material points of masses m1 and m2 and positions and , respectively. The gravitational force exerted by the second body on the first body is

The constant G is the universal gravitational constant and denotes the position of body 2 relative to body 1. Newton’s third law (action and reaction) implies that body 1 exerts an opposite force on body 2: .

Universal Gravitational Constant

The accurate knowledge of the value of G is of considerable importance, both for the computation of space trajectories and for the confirmation of theories such as general relativity.

The first reliable measurement of G was made in 1798 by Henry Cavendish, using a torsion pendulum. The possibility of on-board experiments carried out by satellites has greatly improved the accuracy. The reference value adopted in the international system is: G ≈ 6.674 08 ⋅ 10−11 m3 kg−1 s−2.

Newton’s second law (also called fundamental relation of dynamics) applied to each body in the inertial frame yields the two differential equations

After subtracting the first equality from the second, we obtain the differential equation for the motion of body 2 relative to body 1.

where μ = G(m1 + m2) is called the gravitational constant of the two-body system.

In the restricted two-body problem, the mass of body 2 is negligible compared with that of body 1 (main attracting body). The constant μ = Gm1 is then called the gravitational constant of the main attracting body. This model is well suited to practical applications studying the motion of artificial satellites.

The values of μ for Earth, Moon, and Sun are (in m3/s2):

The solution to the two-body problem is called Keplerian motion, after Johann Kepler, who first set out the laws of motion.

Kepler’s Laws

The laws of planetary motion around the Sun were set out empirically by Johann Kepler (1571–1630) in Astronomia Nova, published in 1609.

The three laws of Kepler are as follows:

• the planets describe ellipses with the Sun as one of their foci;
• the radius vector sweeps equal areas in equal times;
• the square of the period of revolution is proportional to the cube of the semi-major axis.

Kepler had guessed these three laws from the collection of observations made by the Danish astronomer Tycho Brahé (1456–1601) on the motion of the planets.

Their demonstration came much later owing to Isaac Newton (1642–1727).

In Philosophiae Naturalis Principia Mathematica, published in 1687, Newton postulated the laws of motion (inertia, force, and reaction) and the law of gravitation, from which he rigorously established the solution to the two-body problem.

These fundamental results form the basis of orbital and celestial mechanics since the 18th century.

1.2.2 Prime Integrals

A prime integral of a dynamic system is a quantity that remains constant over time. Knowledge of prime integrals helps to delimit the system evolution, even when no analytical solution exists. The two-body problem admits several prime integrals, such as those of angular momentum, energy, and eccentricity vector.

1.2.2.1 Angular Momentum

The angular momentum per unit mass is , where is the velocity of body 2 relative to body 1. Calculating the derivative of with the assumption of a central acceleration given by (1.3) yields

The angular momentum vector is therefore constant throughout the motion.

If and are collinear, then and the motion takes place along a straight line. Such a trajectory is said to be degenerate. Unless otherwise specified, we assume in the sequel that .

As the position and the velocity are constantly perpendicular to , the motion takes place in a plane called the orbital plane.

Taking body 1 as origin O and a reference axis Ox in this plane (Figure 1.1):

• the position of body 2 is defined by the polar angle θ and the radius vector r;
• the velocity of body 2 has radial and orthoradial components and (where denotes the time derivative of x).

Figure 1.1 Orbital plane.

The modulus of the angular momentum is

In an infinitesimal time dt, the radius vector sweeps the area dS of the triangle with sides and . This area is

The areal velocity (area swept per unit time) is therefore constant, just as the angular momentum. This establishes Kepler’s second law or law of areas.

1.2.2.2 Energy

The mechanical energy per unit mass noted w is the sum of the kinetic energy and the potential energy , from which the gravitational force derives.

Taking the scalar product of each member of (1.3) with and using (Figure 1.1), we obtain

Instantaneous variations in kinetic and potential energy cancel each other out. This statement is in fact true for any force deriving from a potential.

The mechanical energy is therefore constant throughout the motion.

The accessible domain depends on the energy sign. To escape the main attracting body, the energy must be zero or positive (kinetic energy ≥ potential energy).

In this case, the escape velocity vesc associated with zero energy (w = 0) and the residual velocity at infinity v∞ (for r → ∞) are respectively given by

1.2.2.3 Eccentricity Vector

The eccentricity vector (also called Laplace vector) is defined as:

To calculate the derivative of , we observe that the radial unit vector , which lies in the orbital plane, rotates around the fixed axis with an angular velocity (see Figure 1.1).

Its derivative is therefore orthoradial and it can be expressed as:

Using (1.5) to substitute , then (1.3) to make appear, we obtain

By deriving equation (1.10), with from (1.4), and using (1.12), the terms cancel each other out and the derivative of is zero.

The eccentricity vector is therefore constant throughout the motion.

1.2.2.4 Useful Relation Between (h, w, e)

By squaring each member of equation (1.10), we obtain the useful relation

This relation can be rewritten by defining two constants p and a from the angular momentum and from the energy respectively, with the convention a = ∞ if w =...