Understanding Orbital Speed and Energy Dynamics

In summary, the total energy of a shuttle in orbit is directly proportional to -1/r. To spiral into a smaller orbit, the shuttle's speed should either increase or decrease, depending on the context. However, it is important to note that the total energy and kinetic energy are equal in value, but the total energy is negative while the kinetic energy is positive. This is significant in understanding the concept of changing orbits.
  • #1
I have a question about orbital speed.
Imagine a shuttle moving in an orbit near the Earth surface,
its total energy is "-GMm/2r", so its total energy is directly proportional to -1/r

In order to spiral into another orbit of smaller radius which mean it would have a larger angular speed, the shuttle should increase or decrease its orginal speeed so that it can get into a lower orbital ?

1) increase
As orbital speed = (GM/r)^1/2, speed should increase in order to get a smaller r.

2) decrease
As total energy directly proportional to -1/r,
and total energy = PE + KE
As r decrease, total energy is more negative and KE should decrease
So the speed should decrease.

Are there anything wrong in these two contradicting concept ?
thank you!
 
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  • #2
Imagine you want put the shuttle in a higher orbit. You must increase its energy. For this, you accelerate the shuttle, which gets more speed (it is still at the same height). Having a bigger speed, the shuttle does not follow the same orbit, but takes an elliptical trajectory with a bigger apogee. In this trajectory the Earth gravitational force brakes the shuttle (do a drawing) and its speed diminishes. At the apogee the energy is always the same but the speed is lower. Now, if you want the shuttle to take a circular orbit with this radius as apogee, you must increment its speed. Otherwise it will follow the elliptical trajectory with the previous perigee.
You cannot just say "to increase the radius you increase speed". To change radius, you must pass through an elliptical trajectory, and then the speed changes.
 
  • #3
lockerman2007 said:
I have a question about orbital speed.
Imagine a shuttle moving in an orbit near the Earth surface,
its total energy is "-GMm/2r", so its total energy is directly proportional to -1/r

In order to spiral into another orbit of smaller radius which mean it would have a larger angular speed, the shuttle should increase or decrease its orginal speeed so that it can get into a lower orbital ?

1) increase
As orbital speed = (GM/r)^1/2, speed should increase in order to get a smaller r.

2) decrease
As total energy directly proportional to -1/r,
and total energy = PE + KE
As r decrease, total energy is more negative and KE should decrease
So the speed should decrease.

Are there anything wrong in these two contradicting concept ?
thank you!

The total kinetic energy is more properly expressed as

[tex]E_t = \frac{GMm}{2a}[/tex]

Where a is the semi-major axis of the orbit( or average radius of the orbit). This allows one to consider elliptical orbits as well.
Total energy can also be expressed as
[tex]E_t = \frac{mv^2}{2}- \frac{GMm}{r}[/tex]
Where r is the radius of your orbit at any given moment.

If you decrease v, then KE drops, The total energy drops, and 'a' decreases. (You drop into an elliptical orbit with a smaller average altitude.)

As you start to "fall" you lose PE. To compensate, your velocity increases to increase your KE. When you reach perigee, your velocity is so large, that you start to climb back out, losing velocity as you do so, unitl you return to the point where you decreased v, and you satrt the cycle all over again.
 
Last edited:
  • #4
lockerman2007 said:
I have a question about orbital speed.
Imagine a shuttle moving in an orbit near the Earth surface,
its total energy is "-GMm/2r", so its total energy is directly proportional to -1/r

In order to spiral into another orbit of smaller radius which mean it would have a larger angular speed, the shuttle should increase or decrease its orginal speeed so that it can get into a lower orbital ?

1) increase
As orbital speed = (GM/r)^1/2, speed should increase in order to get a smaller r.

2) decrease
As total energy directly proportional to -1/r,
and total energy = PE + KE
As r decrease, total energy is more negative and KE should decrease
So the speed should decrease.

Are there anything wrong in these two contradicting concept ?
thank you!
The total energy is negative (-GMm/2r)as you have written. The potential energy also is negative (-GMm/r). The kinetic energy is positive and is equal to +GMm/2r. Therefore, the kinetic energy is greater in an orbit of smaller radius.
Once you note the significance of the negativeness of the potential energy and the total energy, your doubt will be cleared.
Note that the total energy and the kinetic energy are equal in value, but the total energy is negative where as the kinetic energy is positive.
 

1. What is orbital speed and how is it calculated?

Orbital speed refers to the velocity at which an object orbits around another object. It is calculated using the equation v = √(GM/r), where G is the universal gravitational constant, M is the mass of the central object, and r is the distance between the two objects.

2. How does orbital speed affect an object's energy?

Orbital speed directly affects an object's kinetic energy, with a higher speed resulting in a higher kinetic energy. Additionally, an object's potential energy is affected by its distance from the central object, with a greater distance resulting in a higher potential energy.

3. What is the relationship between orbital speed and orbital period?

The relationship between orbital speed and orbital period is described by Kepler's Third Law of Planetary Motion, which states that the square of an object's orbital period is directly proportional to the cube of its average distance from the central object. This means that as orbital speed increases, orbital period decreases.

4. How does an object's mass affect its orbital speed and energy?

An object's mass does not affect its orbital speed or energy, as they are only dependent on the mass of the central object and the distance between the two objects. However, a more massive object will have a stronger gravitational pull, resulting in a higher orbital speed for a smaller object orbiting around it.

5. What factors can cause changes in an object's orbital speed and energy?

Changes in an object's orbital speed and energy can be caused by changes in the mass of the central object, changes in the distance between the two objects, or external forces such as gravitational pulls from other objects or atmospheric drag. Additionally, collisions or interactions with other objects can also cause changes in an object's orbital speed and energy.

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