Semimajor Axis and Eccentricity after increased velocity

In summary, to find the new orbit semimajor axis, eccentricity, and increase in perigee altitude after the space shuttle fires its thrusters at apogee, you can use the vis-viva equation and the equation for perigee altitude. The new semimajor axis can be calculated to be 7809 km, while the new eccentricity and increase in perigee altitude can be determined by solving for these values using the given equations.
  • #1
ryank614
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0
Question:

A space shuttle is in an orbit about the Earth. At its apogee, it uses thrusters and increases its velocity by 400 m/sec. What is the new orbit semimajor axis, eccentricity and how much will the next perigee altitude be increased?

Known:

Original semimajor axis: 7000 km -> a
Original eccentricity: 0.05 -> e
Earth's Radius: 6378 km
u= GxEarth's Mass=3.986x10[tex]^{5}[/tex]

What I have done so far:

I figured out the apogee and perigee of the orbit, as well as the velocity at the apogee before the firing of the thrusters.

i) apogee: a(1+e) = 7350 km
ii) perigee: a(1-e) = 6650 km
iii) velocity at apogee:

[tex]\sqrt{u*((2/r)-(1/a))}/[/tex] where r = apogee.

I got v=7.17 km/s

Now after the thrusters are fired, the new velocity is 7.57 km/s

Using [tex]\epsilon[/tex] = V[tex]^{2}[/tex][tex]/2[/tex] - u[tex]/r[/tex] where r is the current position, aka the apogee and plugging [tex]\epsilon[/tex] into

a = -u[tex]/2\epsilon[/tex]

I found the new semimajor axis to be 7809 km. But then here is the problem. How do I find out the new eccentricity? Thanks!
 
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  • #2




To find the new eccentricity, you can use the vis-viva equation, which relates the velocity of a satellite in orbit to its distance from the center of the Earth. This equation is given by:

v^2 = GM(2/r - 1/a)

Where G is the gravitational constant, M is the mass of the Earth, r is the distance from the center of the Earth, and a is the semimajor axis of the orbit.

Using this equation and plugging in the new velocity (7.57 km/s) and the new semimajor axis (7809 km), you can solve for the new eccentricity.

Additionally, to find the new perigee altitude, you can use the following equation:

r = a(1-e)

Where r is the distance from the center of the Earth at perigee and a is the semimajor axis. Plugging in the new semimajor axis and the new eccentricity, you can solve for the new perigee altitude. Subtracting this value from the original perigee altitude (6650 km), you can determine the increase in the perigee altitude.

I hope this helps! Let me know if you have any further questions.
 
  • #3




Great job on calculating the new semimajor axis and velocity! To find the new eccentricity, we can use the equation e = (r_max - r_min)/(r_max + r_min), where r_max is the new apogee and r_min is the new perigee. We already know the new apogee (a(1+e) = 7809 km), so we just need to find the new perigee.

Since the velocity and semimajor axis have changed, we can no longer use the original equation for the perigee (a(1-e) = 6650 km). Instead, we can use the conservation of angular momentum to find the new perigee. The equation for angular momentum is L = mvr, where m is the mass of the shuttle, v is the velocity, and r is the distance from the center of the Earth.

Since we know the original apogee and velocity, we can calculate the original angular momentum. Then, using the new velocity and semimajor axis, we can solve for the new perigee. Once we have the new perigee, we can plug it into the eccentricity equation and solve for e.

As for the increase in the next perigee altitude, we can simply subtract the original perigee altitude from the new one. Keep in mind that this may not be the exact increase, as the orbit may change slightly due to the firing of the thrusters and the effects of other forces.

I hope this helps and keep up the good work in your calculations!
 

1. What is the semimajor axis after increased velocity?

The semimajor axis is the distance from the center of an orbit to the farthest point, or apogee, of the orbit. After increased velocity, the semimajor axis will remain the same unless there is a significant change in the gravitational pull of the central body.

2. How does velocity affect the semimajor axis?

Velocity has a direct effect on the semimajor axis. An increase in velocity will result in an increase in the semimajor axis, and a decrease in velocity will result in a decrease in the semimajor axis. This is because the orbital path will become wider or narrower as the velocity changes.

3. What is eccentricity and how does it change with increased velocity?

Eccentricity is a measure of how elongated an orbit is. It is calculated by dividing the distance between the two foci of an ellipse by the length of the semimajor axis. Increased velocity can cause a change in eccentricity, as it can either cause the orbit to become more circular or more elliptical depending on the direction of the velocity change.

4. Is there a limit to how much velocity can change the semimajor axis and eccentricity?

Yes, there is a limit to how much velocity can change the semimajor axis and eccentricity. This limit is determined by the strength of the gravitational pull of the central body. If the velocity change is too great, the object may escape the gravitational pull and no longer remain in orbit.

5. How does the change in semimajor axis and eccentricity affect an orbiting object?

The change in semimajor axis and eccentricity can significantly affect an orbiting object. It can change the object's speed, position, and overall stability in its orbit. A change in semimajor axis may also result in a change in the orbital period of the object, while a change in eccentricity can cause the object to come closer or move farther away from the central body at different points in its orbit.

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