An object of mass m orbits a star with mass M , where M >> m . In general the bound or bits are elliptical, which means the kinetic energy of the object T and the gravitational potential energy U may vary. For bound orbits, the total energy E is negative. If the orbit is circular, then T and U are both constant.

To derive the expressions for kinetic energy and gravitational potential energy in a circular orbit, we can start with the equations for these energies.

The kinetic energy (T) of an object is given by the equation:
T = (1/2)mv^2

where:
m is the mass of the object
v is the velocity of the object

The gravitational potential energy (U) is given by the equation:
U = -GMm/r

where:
G is the gravitational constant
M is the mass of the star
r is the distance between the object and the star

In a circular orbit, the velocity of the object can be related to the distance and period of the orbit. The centripetal force required for circular motion is provided by the gravitational force:

F = m(v^2/r) = GMm/r^2

By equating these two forces, we can derive a relationship for the velocity v:

mv^2/r = GMm/r^2

Simplifying, we find:

v^2 = GM/r

Now, let's substitute this result for v^2 into the expression for the kinetic energy T:

T = (1/2)m(GM/r)

Simplifying further, we get:

T = GMm/(2r)

Thus, in a circular orbit, the kinetic energy remains constant.

Now let's calculate the gravitational potential energy U in a circular orbit. Using the equation given earlier and substituting the velocity expression v^2 = GM/r, we find:

U = -GMm/r

This expression indicates that the gravitational potential energy U is also constant in a circular orbit.

In summary, for a circular orbit:
- The kinetic energy T remains constant.
- The gravitational potential energy U also remains constant.

Keep in mind that these results hold assuming a circular orbit. For elliptical orbits, both the kinetic energy and gravitational potential energy vary along the orbit path.

To explain how to get the answer to your question, we'll break it down into a few steps:

Step 1: Understand the scenario
In this scenario, we have an object of mass m orbiting a star with mass M, where M is significantly greater than m. The orbits of the object are generally elliptical, meaning that the object's kinetic energy (T) and gravitational potential energy (U) may vary. For bound orbits, the total energy (E) is negative. Additionally, if the orbit is circular, both T and U remain constant.

Step 2: Determine the energy of the system for bound orbits
To calculate the total energy (E) of the system for bound orbits, we can use the equation:
E = T + U

Since the total energy (E) is negative for bound orbits, it means that the gravitational potential energy (U) must dominate over the kinetic energy (T).

Step 3: Analyze circular orbits
In the case of a circular orbit, we know that the kinetic energy (T) and gravitational potential energy (U) remain constant. This means that the total energy (E) remains constant as well, given that E = T + U. Therefore, for a circular orbit, the total energy (E) will still be negative, as it represents a bound orbit.

Step 4: Apply the understanding
Based on the above analysis, we can conclude that for any bound orbit, whether elliptical or circular, the total energy (E) will be negative. This is due to the dominance of gravitational potential energy (U) over kinetic energy (T) in these orbital systems.

It's important to note that the analysis and equations provided are simplified explanations to help understand the concepts. In more complex scenarios, other factors such as relativity, angular momentum, and eccentricity play crucial roles in determining the characteristics of orbits.