Problem 6 - Astronomy Journey to the Cosmic Frontier

Question

Suppose that a satellite is given a speed \(10 \%\) larger than circular velocity. What would be the shape of the trajectory of the body?

Step-by-Step Solution

Answer

The trajectory of the satellite will be elliptical.

Step 1: Understand Circular Velocity

The circular velocity is the speed that a body must have to maintain a stable circular orbit around a planet. The formula for the circular velocity \(v_c\) around a planet of mass \(M\) at a distance \(r\) is given by \(v_c = \sqrt{\frac{GM}{r}}\), where \(G\) is the gravitational constant.

Step 2: Calculate the Given Speed

Given that the satellite's speed is 10% larger than circular velocity, we denote this speed as \(v = 1.1v_c\). Thus, the new speed is \(v = 1.1\sqrt{\frac{GM}{r}}\).

Step 3: Compare to Escape Velocity

The escape velocity \(v_e\) is the speed needed to break free from a planet's gravitational field, calculated as \(v_e = \sqrt{\frac{2GM}{r}}\). Compare \(v\) with \(v_e\) to determine the orbit type. Notice that \(v < v_e\), since \(1.1v_c = 1.1\sqrt{\frac{GM}{r}} < \sqrt{\frac{2GM}{r}}\), because \(1.1^2 < 2\).

Step 4: Determine Orbit Shape

Since the speed \(v\) is greater than the circular velocity but less than the escape velocity, the orbit of the satellite will be an elliptical trajectory.

Key Concepts

Circular Velocity

In the realm of orbital mechanics, circular velocity is a fundamental concept that describes the perfect speed required for a satellite to remain in a stable circular orbit around a planet. This speed ensures that the gravitational pull of the planet precisely balances the satellite's inertia. The formula for circular velocity \(v_c\) is

• \(v_c = \sqrt{\frac{GM}{r}}\),

where:

• \(G\) stands for the gravitational constant,
• \(M\) symbolizes the mass of the planet, and
• \(r\) is the distance from the center of the planet to the satellite.

This speed allows the satellite to follow a path where its centripetal force, which is the force required to keep it moving in a circle, matches the gravitational pull of the planet.

If a satellite travels at this exact speed, it perfectly follows a circular path maintaining constant altitude and keeps orbiting indefinitely at that radius.

Escape Velocity

Escape velocity is another key concept in orbital mechanics, synonymous with the speed needed for an object to break free from a planet's gravitational influence entirely. It is a critical threshold that allows a spacecraft to travel beyond the gravitational boundaries without falling back to the planet or needing any additional propulsion. The formula for escape velocity \(v_e\) is:

• \(v_e = \sqrt{\frac{2GM}{r}}\).

This formula reveals:

• The stronger the gravitational pull \((G\cdot M)\), the higher the escape velocity.
• The farther from the planet's center \((r)\), the lower this velocity needs to be.

Comparing the speeds, when the satellite's speed is between circular and escape velocities, it will not fly off into space nor stay in a circular path, suggesting another kind of trajectory.

The escape velocity is often, but not always, more than double the circular velocity, since its calculation involves two times the gravitational constant \(G\) compared to just one in the circular velocity.

Elliptical Orbit

An elliptical orbit emerges when an object in space travels at a speed greater than the circular velocity but less than the escape velocity. This creates a beautifully stretched path around a planet. An ellipse is essentially a squashed circle and is one of the conic sections in mathematics. Kepler’s laws of planetary motion heavily center around such orbits. In an elliptical orbit:

• The planet sits at one of the two foci of the ellipse, not the center.
• The satellite's speed varies throughout the orbit, faster at the closest point (periapsis) and slower at the farthest point (apoapsis) due to conservation of angular momentum.

This type of trajectory represents a compromise— the satellite is not bound tightly as in a circular orbit but isn't completely free as it would be at escape velocity. Instead, the satellite traces a predictable yet dynamic pattern, consistently regaining its path due to the ongoing gravitational relationship with the planet.

Elliptical orbits are crucial in understanding many celestial movements, from planets around stars to moons around planets.