Why doesn't the orbit increase in radius?

[IcedB]

The following site shows that when a sail is orientated so that it faces away from the sun, the net force is pushing outwards from the sun and is also opposing the orbit.
http://solarsail.jpl.nasa.gov/introduction/how-sails-work.html

It makes sense that the craft should slow down, but why doesn't the orbit increase in radius?

please help!
IcedB

 

[enigma]

If an object in orbit slows down, by definition its orbit decreases.

The only way to increase the energy of the orbit (the semi-major axis) is to increase the velocity.

 

[IcedB]

http://www.go.ednet.ns.ca/~larry/orbits/kepler.html

v2 = GM/a

so if V increases a (the orbital radius) must decrease. I'm quite confused

 

[BobG]

v^2= GM/a describes the relationship after you've put a satellite into orbit. It doesn't give any insight into how the satellite got into that orbit in the first place.

For any orbit, you have a certain amount of energy. The energy has two components: kinetic energy and potential energy. The kinetic energy comes from the object's motion. The potential energy comes from the gravity.

Kinetic energy is: $$KE=\frac{1}{2}mv^2$$

Potential energy is: $$PE=-\frac{GMm}{r}$$
(The big M is the object you're orbiting. The little m is the orbiting object.)

The potential energy is always negative. In other words, the closer an object is to the Earth, the less negative the value, with a less negative value being greater than a more negative value (pull of gravity is stronger near the Earth than it is thousands of miles out in space).

The total amount of energy depends on the sum of those two and the size of the orbit depends upon the amount of energy in the orbit. If you looked at it by the amount of energy per unit of mass, you'd wind up with:

$$E=\frac{v^2}{2}-\frac{Gm}{r}$$
(the negative value of your potential energy turns your 'sum' into subtraction)

The path of an object depends on the balance between kinetic energy and potential energy. If an object is going fast enough, it will escape the pull of the Earth's gravity and follow a parabolic or hyperbolic trajectory out into the solar system. The value of your total energy determines your trajectory. If your total energy is zero or greater, the object will escape the Earth's gravity (you can rearrange the above equation to find your minimum escape velocity). If the value is negative, the object will follow a closed orbit (an ellipse or circle).

The size of the orbit depends on the energy of the orbit:

$$a=-\frac{GM}{2E}$$

Since potential energy is negative, the more energy the orbit has (the less negative the value), the bigger the semi-major axis (average radius). The less the energy (the more negative the value), the smaller the orbit.

Unless you can instantly teleport an object from one location to another, it's going to be pretty hard to change the value of your potential energy for any given instant. You can change your kinetic energy, though. If you increase the velocity, you increase the kinetic energy, pushing the total energy less negative. If you decrease the kinetic energy, you push the total energy more negative, decreasing the size of the orbit.

You can't think of it as increasing the satellite's speed. Instead, you're increasing the amount of energy in the orbit to get it to a higher orbit. Even though you may be adding kinetic energy, energy is energy and it will be distributed between kinetic and potential (in other words, if you move further from the Earth, the potential energy has to become more negative and the only way to do that is to make your kinetic energy less positive, since the total can't change without some external force).

So, your equation, v^2=GM/a, only describes the relationship that exists when you get to your final orbit. It's describing the final relationship between a satellite's kinetic energy and potential energy. It doesn't explain how you were able to get the satellite from Earth to it's final orbit, or the process you used to add enough energy to get the satellite there.

 

[Everett]

Here is another way to look at this. In the relevant diagram at the URL you give, the sail diverts solar flux 90 degrees so that it is directed along the direction of motion of the satellite. Diverting the solar flux requires an exchange of momentum; there is a force on the satellite where the radial component is directed outward and the tangential component is directed opposite the motion of the satellite.

As you state, the tagential velocity must decrease, though very slowly. The tangential acceleration from the sail is very small, but it has a cumulative affect on the velocity of the satellite. However, as the velocity decreases, then the satellite is moving too slowly to compensate for the inward acceleration from solar gravity, so the radius of the orbit begins to decrease. For the circular orbit, GM/r^2 equaled v^2/r. With v decreasing, then the first term (solar gravity) becomes larger than the second. The satellite begins to accelerate radially toward the Sun.

But what about that component of force from the sail that is directed radially outward? It is perpendicular to the tangential velocity, so it has no effect there. It does slightly subtract from the solar gravitational force, but only by a small, unchanging amount for a given radius. It would make the radius shift outward slightly, initially, if nothing else was going on. However, it cannot compete with the continual decrease of tangential velocity due to the tangential acceleration from the sail. At the extreme, the tangential acceleration would eventually decrease the tangential velocity to zero, and then the satellite would fall straight into the Sun!

You do need to be careful about terms like "speeding up" or "slowing down" for orbits. For the situation that we are discussing, intially, the velocity does decrease. However, as the satellite moves out of its circular orbit and accelerates toward the Sun, then both its angular and its linear velocity would actually increase!