How to calculate the required force necessary for orbit

In summary: It decreases as the orbital distance gets closer to the center of the planet. For a given force, an object in a circular orbit will move in a smaller and smaller circle until it reaches the center of the planet.
  • #1
Ameise
3
0
Hello,

I am working on a software application which will require me to generate objects in orbit around other objects.

Now, this is my belief, and I am asking for correction if I am wrong---

If given two objects, O1 and O2, with masses M1 and M2, and I want to make O2 orbit O1 at an orbit at distance r with eccentricity e, will this work:

force_for_orbit = (G * ((M1 * M2) / r^2)) * (e + 1)
velocity = force_for_orbit / M2

and the vector for velocity would need to be a perfect right angle (either direction) from the vector for the direction of the gravitational force towards M1, or in other words, the vector for velocity would need to lie tangent to the curve that would be generated by the orbit?

My belief is that a force that is equal to the force being applied by gravity but being applied 90 degrees opposing to it will cause the object to move in a circle, IE with eccentricity of 0. Because multiplying by 0 will not work, I add to make it work... if e becomes greater and greater it will still work until > 1 which escapes gravity.

Thank you!
 
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  • #2
Judging by your nickname you speak German. This might help you:
http://de.wikipedia.org/wiki/Kosmische_Geschwindigkeiten

To get a circular orbit (e = 0) you need a speed (perpendicular to radial direction):
[tex]v_1 = \sqrt{\frac{ G M_1 }{r}}[/tex]

For an elipse (0 < e < 1):
[tex]v_1 < v < v_1 \sqrt{2}[/tex]
 
  • #3
Ameise said:
Hello,

I am working on a software application which will require me to generate objects in orbit around other objects.

Now, this is my belief, and I am asking for correction if I am wrong---

If given two objects, O1 and O2, with masses M1 and M2, and I want to make O2 orbit O1 at an orbit at distance r with eccentricity e, will this work:

force_for_orbit = (G * ((M1 * M2) / r^2)) * (e + 1)
velocity = force_for_orbit / M2

The Newtonian gravitational force between two objects is always [itex]GM_1M_2/r^2[itex]. The eccentricity of the orbit is not a part of the equation. Neither is the velocity.

To make an object orbit with some non-zero eccentricity you need to give it an initial velocity that either (a) has a non-zero component parallel to the radial vector, or (b) has a magnitude different from the circular orbit velocity.
 
  • #4
The reason that I added eccentricity was that I am trying to form a linear equation in which I can set the parameters for mass AND eccentricity and get a result that my application can use. Being that it is technically a 3d simulation, all of my work is done using vectors (technically matrices, but vectors would be the basis of velocity), and the velocity would obviously be (force / mass) * direction_vector.

If I applied my equation, what would be the result of it?

My equation roughly becomes when written better:

gravity = (G * ((M1 * M2) / r*r))
force = gravity * (e + 1) //not proper physics, but I think that it should generate the proper effect?
velocity_vector = right_angle(gravity_Vector) * force / M2
 
  • #5
One issue that crops up is that a force changes both the shape and the average radius of an orbit. It's more efficient if the direction of the force is perpendicular to gravity, using velocity changes to change the total energy (kinetic plus gravitational potential) and path. You'll need at least two bursts, the first one creating an ellipitcal orbit with some amount of kinetic and gravitational potential energy, the second one adjusting the shape of the orbit as well as making the final total energy adjustment.

There is math to optimize the required force (or fuel) to get from one circular orbit to another circular orbit:

http://en.wikipedia.org/wiki/Hohmann_transfer_orbit
 
Last edited:
  • #6
Ameise said:
The reason that I added eccentricity was that I am trying to form a linear equation in which I can set the parameters for mass AND eccentricity and get a result that my application can use.
But what does it need as the result? The start velocity perpendicular to radial direction? You could try linear interpolation with the formulas from post #2:

[tex]v = (1 + e(\sqrt{2} - 1)) \sqrt{\frac{ G M_1 }{r}}[/tex]

I doubt that it is that simple. The eccentricity is probably not linear to the initial velocity. But if it just has to look good... :smile:
 
  • #7
What are you trying to accomplish, Ameise? If you are trying to build a realistic 3DOF simulation you need to use a realistic model of gravity: Newton's law of gravity is pretty close to correct for most planetary applications. In Newtonian mechanics, the gravitational force depends on distance and nothing else. In an elliptical orbit the force is not constant.
 

1. How do I calculate the required force for orbit?

The required force for orbit can be calculated using Newton's Law of Gravitation, which states that the force of gravity between two objects is equal to the product of their masses divided by the square of the distance between them. In the case of orbit, the force required to keep an object in orbit around a larger object (such as a planet or star) can be calculated by equating this force of gravity to the centripetal force needed to maintain circular motion.

2. What factors affect the required force for orbit?

The required force for orbit is affected by the mass of the object being orbited, the distance between the two objects, and the velocity of the object in orbit. The greater the mass of the object being orbited, the greater the force of gravity and therefore the greater the required force for orbit. Similarly, the closer the distance between the two objects, the greater the force of gravity and the greater the required force for orbit. Finally, the faster the velocity of the object in orbit, the greater the centripetal force required to maintain that orbit.

3. How can I calculate the velocity required for orbit?

The velocity required for orbit can be calculated using the formula v = √(GM/r), where G is the universal gravitational constant, M is the mass of the larger object, and r is the distance between the two objects. This formula assumes a circular orbit and is derived from equating the force of gravity to the centripetal force. Note that this formula gives the minimum velocity required for orbit, and the actual velocity may be higher depending on the shape and size of the orbit.

4. Is the required force for orbit different for different types of orbits?

Yes, the required force for orbit can vary depending on the type of orbit. For example, a circular orbit requires a different amount of force compared to an elliptical orbit. Similarly, a geostationary orbit (where the orbiting object remains above a fixed point on the planet's surface) requires a different amount of force compared to a polar orbit (where the orbiting object passes over the planet's poles). The required force for orbit will also differ based on the altitude of the orbit.

5. Can the required force for orbit be calculated for any two objects?

Yes, the required force for orbit can be calculated for any two objects as long as their masses and the distance between them are known. This calculation can be used for any objects in orbit, whether it is a planet orbiting a star, a satellite orbiting a planet, or two spacecraft orbiting each other. However, this calculation assumes that the objects are point masses and does not take into account any external forces such as atmospheric drag or gravitational influences from other objects.

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