What Is the Eccentricity of an Orbit Given Vmin and Vmax?

In summary, to find the eccentricity of a satellite's orbit around Earth with velocities between Vmin=V-V0 and Vmax=V+V0, you can use the conservation of angular momentum. By relating the speed at aphelion to the speed at perihelion, you can get a simple expression for the eccentricity in terms of V and V0. This can be achieved by using the equation rmin=l^2/(Gm1m2*mu*(1+e)) and the fact that rmin = a(1-e) and rmax = a(1+e).
  • #1
HHveluj
5
0

Homework Statement


A satellite in an orbit around Earth has velocities between Vmin=V-V0 and Vmax=V+V0. Find the eccentricity of the orbit.

Homework Equations


rmin=l^2/(Gm1m2*mu*(1+e))
E=(Gm1m2)^2*mu*(e^2-1)/2l^2
where mu = reduced mass = m1m2/m1+m2

The Attempt at a Solution


I know all this formulas for eccentricity, but they all involve disctances/or energy, but not velocities! I don't even know how to start solving this... Any help/hints?

Thank you very much in advance!
 
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  • #2
HHveluj said:

Homework Statement


A satellite in an orbit around Earth has velocities between Vmin=V-V0 and Vmax=V+V0. Find the eccentricity of the orbit.

Homework Equations


rmin=l^2/(Gm1m2*mu*(1+e))
E=(Gm1m2)^2*mu*(e^2-1)/2l^2
where mu = reduced mass = m1m2/m1+m2

The Attempt at a Solution


I know all this formulas for eccentricity, but they all involve disctances/or energy, but not velocities! I don't even know how to start solving this... Any help/hints?

Thank you very much in advance!


You may use conservation of angular momentum to relate the speed at aphelion to the speed at perihelion. Recall, [itex] {\vec L} = {\vec r} \times {\vec p} [/itex]. Since at aphelion and at perihelion the motion is perpendicular to the position vector, you get that [itex] r_{min} v_{max} = r_{max} v_{min} [/itex]. Since [itex] r_{min} = a(1-e) [/itex] (*if* I recall correctly) and [itex] r_{max} = a(1+e) [/itex] , you get a simple equation relating the max and minimum speeds. You will get a simple expression for the eccentricity in terms of V and V_0.

Hope this helps.

Patrick
 

Related to What Is the Eccentricity of an Orbit Given Vmin and Vmax?

1. What is a satellite in orbit around Earth?

A satellite in orbit around Earth is an object that is placed into orbit around the Earth by a rocket. It is used for a variety of purposes, such as communication, navigation, and scientific research.

2. How does a satellite stay in orbit around Earth?

A satellite stays in orbit around Earth due to the balance between its forward momentum and the pull of Earth's gravity. This allows the satellite to continuously orbit the Earth without falling back to the surface.

3. How fast does a satellite travel in orbit around Earth?

The speed of a satellite in orbit around Earth depends on its altitude. A satellite in low Earth orbit travels at a speed of around 17,500 miles per hour, while a satellite in geostationary orbit travels at a speed of around 6,800 miles per hour.

4. How do scientists control a satellite in orbit around Earth?

Scientists can control a satellite in orbit around Earth through ground stations that send signals to the satellite, allowing them to adjust its position, orientation, and other functions. They can also upload commands and receive data from the satellite through these ground stations.

5. What are some benefits of having satellites in orbit around Earth?

There are numerous benefits of having satellites in orbit around Earth. These include providing communication services, monitoring weather patterns, aiding in navigation, conducting scientific research, and supporting military operations. Satellites also help us gain a better understanding of our planet and the universe.

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