Orbital Mechanics Part 2
So, an object is in orbit around a planet. How fast does it need to go for that orbit to be circular?
Photo Credit: Me
Let’s imagine the planet has a mass of M and the object has a mass of m.
Let’s also imagine that M≫m. The distance between the two centre of masses is r. The velocity of the object is v, and as we are going to be considering circular orbits only, v is at right angles to r.
If v is not just right, or the angle is not 90 degrees, then we have an elliptical orbit, the idea is similar, but a bit trickier to deal with.
As the object is moving in a circle, the force toward the centre of the circle is given by this equation. I hope to derive this for you in a future post.
The gravitational force between the two masses is given by this equation. G is the universal gravitational constant. This is the Newtonian Law of Gravitation, and in I will show how this can be found from first principles in part 3 of this series.
We can equate these two expressions, and thus link the radius of a circular orbit to the velocity needed for that orbit to be circular.
We can link the velocity with the circumference of the orbit and the time it takes to orbit.
As we now have two expressions with ‘v’, they can be combined, and thus the duration of the orbit is linked to the radius of the circular orbit, and we get to Keppler’s law, where the period squared is proportional to the cube of the radius of orbit (k is a constant).
What if the speed is not just right? Let us imagine the speed is a little different from that given by the the equation we worked out above equation (which gives the speed needed for circular motion).
If the speed is a little too high, the curvature will be lower, and the distance will increase. The object slows, making the path more curved. Eventually the speed is low enough and the curvature pronounced enough that the object begins to move toward the planet again. The path will be elliptical (the exact proof of this is trickier).
If the speed is too low, the curvature is more pronounced and the distance reduces. As the object has some motion toward the planet, it speeds up.
This makes the path less curved. Eventually the speed is big enough and the curvature has reduced enough so the object moves away from the planet again.
This picture shows an elliptical path (the red line). Where it touches the thick dashed line, it is too slow to move in the circular orbit described by the thick line, so it falls back toward the planet.
Where it touches the thin dashed line it is moving too fast to maintain that circular orbit, and so the path is less curved than the circle and it moves away from the planet.
The speed has to be just right for circular orbits.
What’s Next?
We now have everything we need to think about some of the basics of orbital manouvering, that will be Part 4.
Part 3 will be a justification of the Newtonian law of gravity (used above)
Links
Title image via this page. (public domain)
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