Question

What would become of the earth's orbit (which you may consider to be a circle) if half of the sun's mass were suddenly to disappear? Would the earth remain bound to the sun? [Hints: Consider what happens to the earth's KE and PE at the moment of the great disappearance. The virial theorem for the circular orbit (Problem 4.41) helps with this one.] Treat the sun (or what remains of it) as fixed.

Short answer

Earth's orbit becomes elliptical, but it remains bound to the sun.

Step-by-step solution

Understanding the Earth's Initial Orbit

Initially, the earth orbits the sun in a nearly circular path. The gravitational force provides the necessary centripetal force for this circular motion. According to the virial theorem, for a stable circular orbit, the kinetic energy (KE) is related to the gravitational potential energy (PE) by \( KE = -\frac{1}{2} PE \).

Analyze the Changes in Gravitational Force

When half of the sun's mass suddenly disappears, the gravitational force experienced by the earth is halved. The new gravitational force \( F' \) becomes \( F' = \frac{G \cdot (M/2) \cdot m}{r^2} \), where \( M \) is the original mass of the sun, \( m \) is the earth's mass, and \( r \) is the radius of the orbit.

Change in Potential Energy (PE')

The potential energy in the gravitational field is given by \( PE = -\frac{G \cdot M \cdot m}{r} \). After the mass disappears, the potential energy becomes \( PE' = -\frac{G \cdot (M/2) \cdot m}{r} = \frac{1}{2} PE \).

Conservation of Kinetic Energy (KE)

The kinetic energy of the earth does not change instantly when the mass of the sun changes because KE depends on the velocity of the earth which remains constant immediately after mass reduction. Thus, immediately after the disappearance, \( KE' = KE \).

Determine the New Nature of Earth's Orbit

With the sudden loss of mass, the gravitational attraction drops, affecting the force balance necessary for circular motion. As the centripetal force decreases and KE remains initially unchanged, the orbit becomes elliptical, with the earth possessing greater kinetic energy than required for the new, smaller gravitational pull, leading the earth to move away from the sun.

Key concepts

Gravitational Potential Energy

Gravitational potential energy (PE) is the energy an object possesses due to its position in a gravitational field. In the context of celestial bodies, it is essential for understanding how objects like planets and stars interact. For a planet orbiting nearly circular around the sun, the gravitational potential energy is determined by the equation: \[ PE = -\frac{G \cdot M \cdot m}{r} \]where:- \( G \) is the gravitational constant,- \( M \) is the mass of the sun,- \( m \) is the mass of the Earth,- \( r \) is the distance between the Earth and the Sun.This formula shows that potential energy is negative, indicating that an energy input would be needed to separate the two bodies. After half of the sun's mass disappears, the gravitational potential energy reduces to half its original value, aligning with the new mass \( M/2 \). This change is crucial because it alters the force that maintains the Earth's orbit.

Kinetic Energy

Kinetic Energy (KE) refers to the energy an object possesses due to its motion. In the scenario of Earth's orbit around the sun, the kinetic energy is derived from Earth's velocity as it moves through space. The formula for kinetic energy is:\[ KE = \frac{1}{2} mv^2 \]where:- \( m \) is the mass of the Earth,- \( v \) is the velocity of the Earth in its orbit.Immediately after a drastic change like the reduction of the Sun’s mass, Earth's velocity remains constant for an instant, meaning its kinetic energy stays the same irrespective of the sun's mass change. This unchanging kinetic energy becomes significant when the gravitational force diminishes. It results in an orbit shift because the remaining gravitational pull isn't enough for a circular path, causing the Earth to transition into a more elliptical orbit.

Virial Theorem

The virial theorem is a key principle in astrophysics that relates the average total kinetic energy and the average total potential energy of a stable, bound system in equilibrium. For planets in stable orbit, the theorem is expressed as:\[ 2 \langle KE \rangle = -\langle PE \rangle \]Simplifying, for a circular orbit:\[ KE = -\frac{1}{2} PE \]This relationship implies that, in a stable orbit, the planet's kinetic energy is half the absolute value of its potential energy. When half of the sun’s mass vanishes, this delicate balance is disrupted. The gravitational potential energy drops, but the kinetic energy does not instantly change, disturbing the initial equilibrium. The Virial theorem explains why Earth would not maintain its circular orbit when the gravitational potential energy drops; the balance required for a circular trajectory is lost, causing the Earth to start moving in an elliptical path.