Problem 20 - Classical Mechanics - An Undergraduate Text

Question

A body moving in an inverse square attractive field traverses an elliptical orbit with major axis \(2 a\). Show that the time average of the potential energy \(V=-\gamma / r\) is \(-\gamma / a\). [Transform the time integral to an integral with repect to the eccentric angle \(\psi\).] Deduce the time average of the kinetic energy in the same orbit.

Step-by-Step Solution

Answer

Therefore, the time average of the potential energy in an inverse square attractive field with an elliptical orbit is \(-\gamma / a\), and the time average of the kinetic energy is \(-\gamma / 2a\).

Step 1: Start with the given statement

Start from the given potential energy \(V=-\gamma / r\), which represents the motion energy according to the distance from the central body \(r\).

Step 2: Convert time integral to eccentric angle integral

Applying the given hint to transform the time integral into an integral with respect to the eccentric angle \(\psi\), consider Kepler's second law, which implies that equal areas are swept out in equal times. This provides us with the derivative of the eccentric anomaly with respect to time, which in turn gives us the relationship needed to transform the integral. Thus, \(\frac{dt}{d\psi}=\frac{(1 - e \cos(\psi))^2}{(1 - e^2)^{3/2}} a^{3/2} \), where \(e\) is the eccentricity of the elliptical orbit.

Step 3: Calculate time average of potential energy

The time average of potential energy can be represented as an integral over one period \(T\), \(\frac{1}{T}\int_0^{T} -\gamma/r dt\). Using the relationship derived in the previous step to change the variable of integration from \(t\) to \(\psi\), and given that for an elliptical orbit \(r=a(1-e\cos(\psi))\), the integral becomes \(\frac{-\gamma}{2 \pi a (1 - e^2)^{3/2}} \int_0^{2\pi} \frac{(1 - e \cos(\psi))^2}{1 - e \cos(\psi)} d\psi = -\gamma/a\). Hence, the time average of potential energy in this inverse square attractive field is \(-\gamma/a\).

Step 4: Deduce time average of kinetic energy

Knowing that the total mechanical energy \(E\) in an elliptical orbit is conserved and equals half the time-averaged potential energy, \(E = 1/2(V)_{avg} = -\gamma/2a\), and considering that the total energy is the sum of kinetic and potential energies, \(E = (T)_{avg} + (V)_{avg}\), where \((T)_{avg}\) is the time averaged kinetic energy, we therefore deduce that \((T)_{avg} = -\gamma/2a\).

Key Concepts

Potential Energy in Orbital Mechanics

Understanding the role of potential energy in orbital mechanics is essential to comprehend how celestial bodies move in space. In the realm of such mechanics, we often consider the force of gravity as an inverse square law. This means that the force exerted by a mass, such as a planet or star, decreases with the square of the distance from that mass. Potential energy \( V \) in this context is given by the formula \( V = -\gamma / r \), where \( \gamma \) is a constant related to the masses involved and the gravitational constant, and \( r \) is the distance to the center of the mass.

For a body in orbit, this potential energy ensures that the body follows a trajectory around the mass, rather than flying off straight. It is this balance between the gravitational pull (potential energy) and the body's inertia (kinetic energy) that creates the conditions for an orbit to exist.

Elliptical Orbit Characteristics

Elliptical orbits are the most common type of orbit and are characterized by their shape, which resembles an elongated circle. One of the unique features of an elliptical orbit is that it has two foci, and the central celestial body, such as the Sun in our Solar System, is located at one focus, not the center. Two primary measurements describe the size and shape of an ellipse: the major axis is the longest diameter, and the eccentricity describes how stretched out the ellipse is.

With orbits, the distance between the orbiting body and the central mass changes as the body moves along its path. The point closest to the central body is termed the periapsis, and the farthest point is the apoapsis. These variations in distance result in changes in the gravitational potential energy and the speed of the orbiting body – it moves fastest at the periapsis due to its maximal kinetic energy and slowest at the apoapsis due to maximal potential energy.

Kepler's Laws of Motion

Johannes Kepler's laws of motion provide a succinct description of the motion of planets and other celestial bodies in space. The first law, sometimes called the law of orbits, states that planets move in elliptical orbits with the Sun at one focus. This law aligns with the elliptical orbit characteristics and implies that the orbits are not perfect circles.

The second law, known as the law of areas, says that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This means that planets move faster when they are nearer to the Sun and slower when they are further away, an aspect crucial for transforming integrals in orbital mechanics. Kepler's third law, the law of periods, relates to the time a planet takes to orbit the Sun, stating that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. These laws are foundational for understanding the gravitational interactions between celestial bodies and their resultant motion.

Mechanical Energy Conservation

The conservation of mechanical energy is a principle stating that the total mechanical energy in a system remains constant if there are no losses or additions of energy due to external forces. In the context of orbital mechanics, this concept is particularly significant as it dictates the relationship between potential and kinetic energies for an object in motion, like a satellite or planet in an elliptical orbit.

When a body moves in an elliptical orbit under the influence of an inverse square law force, like gravity, the mechanical energy is conserved because the only forces at work are conservative in nature. The total mechanical energy is a sum of kinetic energy, which is the energy due to the object's motion, and potential energy, which is the energy due to the object's position in a gravitational field. For an orbiting body, as potential energy decreases (when moving closer to the central mass), kinetic energy increases, and vice versa, but their sum remains constant – this is the principle of mechanical energy conservation at work in an elliptical orbit.