Orbital energy of ellipse and hyperbolic trajectory

• IPhO' 2008
In summary, the orbital energy of an ellipse is the total energy required for an object to maintain a stable, elliptical orbit around another object. It is calculated using the formula E = -G(m1m2)/2a and determines the shape and stability of an object's orbit. The orbital energy of a hyperbolic trajectory is positive, indicating that the object has enough energy to escape its orbit, while that of an ellipse is negative. The orbital energy can be changed through various means, but this can have significant effects on an object's orbit and trajectory.
IPhO' 2008
Please tell me how to find the orbital energy of ellipse and hyperbolic trajectory.
Thank you.

In both cases, it is the sum of the kinetic energy and gravitational potential at any point of the trajectory.

Alternatively, if you know the semi-major axis(a), it can be found by:

$$-\frac{GM}{2a}$$
for an ellipse

and

$$\frac{GM}{2a}$$ for a hyperbola.

Please show me how to prove it.

1. What is the orbital energy of an ellipse?

The orbital energy of an ellipse refers to the total energy required for an object to maintain a stable, elliptical orbit around another object. This energy is a combination of the object's kinetic energy (due to its motion) and its potential energy (due to its position in the gravitational field).

2. How is the orbital energy of an ellipse calculated?

The orbital energy of an ellipse can be calculated using the formula E = -G(m1m2)/2a, where G is the gravitational constant, m1 and m2 are the masses of the two objects, and a is the semi-major axis of the ellipse. This equation takes into account the effects of both the object's kinetic and potential energy.

3. What is the significance of the orbital energy of an ellipse?

The orbital energy of an ellipse is significant because it determines the shape and stability of an object's orbit. Objects with lower orbital energy will have more circular orbits, while those with higher orbital energy will have more elliptical orbits. Additionally, the orbital energy can also affect the speed and trajectory of an object as it moves through its orbit.

4. How does the orbital energy of a hyperbolic trajectory differ from that of an ellipse?

The orbital energy of a hyperbolic trajectory is positive, while that of an ellipse is negative. This means that objects on a hyperbolic trajectory have enough energy to escape the gravitational pull of the object they are orbiting, while those on an elliptical orbit do not. Hyperbolic trajectories are typically associated with comets and other objects in highly eccentric orbits.

5. Can the orbital energy of an ellipse or hyperbolic trajectory be changed?

Yes, the orbital energy can be changed through various means such as thrust from a spacecraft's engines, gravitational assists from other objects, or interactions with other objects in the orbit. However, changes in orbital energy can also have significant effects on an object's orbit and trajectory, so they must be carefully planned and executed.

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