# Circular orbit change after gaining mass ejected from sun

• caitphys
In summary, by considering the angular momentum and circular motion of the satellite, its initial angular momentum and energy can be calculated. With this information, the initial orbit of the satellite can be determined.
caitphys

## Homework Statement

A satellite of mass m is in a circular orbit of radius R about a star of mass M. The star ejects 1% of its mass by means of a spherically symmetric wind which removes the mass to a large distance. What are the new nearest and furthest distances of the satellite’s orbit around the star?

## Homework Equations

Angular momentum $$\vec L=\vec r \times \vec p$$

Total energy of orbit $$E= \frac {1}{2}m\dot r^2+\frac{L^2}{2mr^2} -\frac{GMm}{r}$$

surface area of a sphere $$S=4\pi r^2$$

## The Attempt at a Solution

If m<<M then it can treated as a point source. So the satellite gains mass ##\frac{M}{400\pi R^2}##

The angular momentum of the circular orbit of the satellite ##\vec L = mr \times \dot r = mr \times (\vec\omega \times \vec r)=mr^2\omega##

For circular motion ##E=\frac{1}{2}mv_0^2 - \frac{GMm}{R}##

I am not entirely sure how to approach this question, as when I considered it I thought that angular momentum and energy should both increase, as the satellite gains mass?

Either way, I calculated the initial angular momentum and energy for the circular orbit of the satellite:

##L=mr^2\omega## and ##\omega=\frac{v_0}{r}## so ##L_0=\frac{GMm}{v_0}##

Now as ##\dot r## is zero for a circular orbit, ##E_0=\frac{1}{2}mv_0^2 - \frac{GMm}{r}##, and equating gravitational force with that of circular motion, ##R=\frac{GM}{v_0^2}## so ##E_0=\frac{1}{2}mv_0^2##.

Now I guess it could be argued that angular momentum is constant as it is conserved for radial forces, but even so, I don't really see how this would help me equate energies or something, as if energy does change, I'm not sure by which factor? Equally, I suppose potentially energy could be conserved if its seen as the total energy of the sun/satellite system, but I really am at a bit of a loss.

I'd really appreciate a bit of a steer in a more productive direction, thanks!

You know the initial velocity and position of the satellite. This should be sufficient to find its orbit. There is no need to look at energies.

## 1. How does gaining mass ejected from the sun affect an object's circular orbit?

When an object gains mass ejected from the sun, its circular orbit will increase in size. This is because the added mass increases the gravitational pull on the object, causing it to move further away from the sun.

## 2. Does the change in circular orbit depend on the amount of mass ejected from the sun?

Yes, the change in circular orbit does depend on the amount of mass ejected from the sun. The more mass that is ejected, the greater the increase in the object's circular orbit will be.

## 3. Can gaining mass ejected from the sun cause an object's circular orbit to become elliptical?

Yes, gaining mass ejected from the sun can cause an object's circular orbit to become elliptical. This is because the added mass can change the balance of forces acting on the object, resulting in a more elongated orbit.

## 4. How does the velocity of an object change after gaining mass ejected from the sun?

The velocity of an object will decrease as it gains mass ejected from the sun. This is due to the conservation of angular momentum, which states that as the radius of an object's orbit increases, its velocity decreases.

## 5. Is it possible for an object's circular orbit to remain unchanged after gaining mass ejected from the sun?

No, it is not possible for an object's circular orbit to remain unchanged after gaining mass ejected from the sun. The added mass will always cause some change in the object's orbit, even if it is minimal.

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