Apparent contradiction with orbital mechanics

[ee78]

Consider a two body system for simplicity: M1 << M2. M1 is in a circular orbit about M2.

The tangential speed of M1 is inversely proportional to its distance from the center of M2. However, in order to move to a lower orbit, or to de-orbit completely, M1 must lose tangential speed (in the case of an artificial satellite, the application of reverse thrusters is the most common method), and to go to a higher orbit, M1 must gain tangential speed. So what happens in the case of moving to a lower orbit? M1 briefly slows down, but then immediately gains speed as it falls into a lower orbit?

 

[Janus]

As the satellite loses speed, it begins to fall in towards the Earth. As it does so, it begins to pick up speed due to its change in gravitational potential. At some point its speed will equal that needed to maintain a new lower circular orbit for that altitude, but the direction of its velocity will still have it moving in the downward direction, and it will "overshoot" this altitude. It will continue to fall and gain speed until it reaches a point where the tangential speed is enough to stop its inward fall. By this time, its speed is much greater than that needed to maintain an orbit at that altitude. So, it will begin to climb back up, losing speed as it does so, until it returns to the point where it initially lost speed. It then repeats the process. IOW, the satellite enters a elliptical orbit with a new lower perigee and its starting altitude as the apogee.

If you want the satellite in a lower circular orbit, you have to fire the thrusters again at perigee to shed speed until it matches that of a circular orbit for that altitude.

 

[ee78]

Thank you for your response. I have three questions:

1) In other words, the point in the circular orbit where reverse thrusters are applied (assume a reverse thrust vs. time curve that mimics a delta function for simplicity, so that the reverse thrust is intense, yet very brief, so that we may talk about a "point of applicaton") becomes the apogee of the new elliptical orbit?

2) Say the satellite already is in an elliptical orbit, and ground control wants the satellite to enter a new elliptical orbit that has both an apogee and perigee closer to the center of the Earth than the old orbit's apogee and perigee (in other words, a new elliptical orbit completely inside the old elliptical orbit), how would this be accomplished?

3) Finally, what would respective graphs of tangential velocity versus time and radial velocity versus time look like for an artificial satellite undergoing long-term, continuous orbital decay due to upper atmospheric drag?

 

[Janus]

Thank you for your response. I have three questions:

1) In other words, the point in the circular orbit where reverse thrusters are applied (assume a reverse thrust vs. time curve that mimics a delta function for simplicity, so that the reverse thrust is intense, yet very brief, so that we may talk about a "point of applicaton") becomes the apogee of the new elliptical orbit?

Yes

2) Say the satellite already is in an elliptical orbit, and ground control wants the satellite to enter a new elliptical orbit that has both an apogee and perigee closer to the center of the Earth than the old orbit's apogee and perigee (in other words, a new elliptical orbit completely inside the old elliptical orbit), how would this be accomplished?

Wait until the satellite reaches apogee. Apply a delta V which moves it into an orbit with the lower perigee (and the same apogee). Wait until the satellite reaches the new perigee and apply a delta which changes the orbit to one with a lower apogee. As a rule, Changes in velocity made at apogee change the perigee, and changes made at perigee change the apogee.

3) Finally, what would respective graphs of tangential velocity versus time and radial velocity versus time look like for an artificial satellite undergoing long-term, continuous orbital decay due to upper atmospheric drag?

Both will increase with time (assuming we started with a circular orbit.) The tangential velocity will not ever increae enough to cancel the Fall.

 

[ee78]

Both will increase with time (assuming we started with a circular orbit.) The tangential velocity will not ever increae enough to cancel the Fall.

1) Interesting. Is there any other known example of a drag force actually increasing the velocity of an object?

For the case of a circular orbit at the onset of orbital decay, I imagine a steady decrease in tangential velocity, and a steady increase in radial velocity, as the satellite spirals inward. How can the tangential velocity increase when the only force acting in the tangential direction is air resistance (unless gravity is somehow causing both velocity components to continually increase)?

Put another way, I am having difficulty visualizing both components of velocity increasing throughout the entire process because of causality. Obviously, as the satellite loses altitude, it loses potential energy. Some of this lost potential energy goes to increasing the kinetic energy as the satellite accelerates, and some is lost to heating the surrounding air; thus, the satellite's total energy (kinetic plus potential) is continually decreasing. It's just hard to wrap my mind around the concept of a drag force actually causing an object to gain speed, with no prior temporary decrease in speed. Shouldn't the drag force directly sap the object of its kinetic energy, and thereby indirectly sap the object of potential energy after that?

2) Will both velocities increase monotonically with time if the satellite started in an elliptical orbit?

For the case of gradual, long-term orbital decay, I don't see how the answer can be "yes". I do realize that technically, the satellite does not move in a closed curve throughout this process, though each orbital cycle should be a close approximation to an ellipse. Therefore, the increases (approaching "perigee") and decreases (approaching "apogee") in tangential velocity, as well as increases and decreses in radial velocity (with the radial velocity obviously being zero at both "apogee" and "perigee") should still occur in this scenario. Am I correct?

Note: I used quotation marks for apogee and perigee because, as noted, the trajectory would no longer be a closed path.

 

[D H]

Suppose you have a satellite orbiting in the exoatmosphere. There will be more drag at perigee than apogee if the orbit is non-circular. This acts to circularize the orbit. The orbital velocity will increase as the satellite falls. This increases the drag on the satellite because it is moving faster (drag is roughly proportional to the square of the speed at these high velocities) and because it is plowing ever deeper into the atmosphere.

The velocity increases so long as atmospheric drag is just a perturbation effect. However, eventually the satellite will have plowed so deep into the atmosphere that drag becomes substantial. The satellite will then slow down quite quickly, just before it burns up.

 

[D H]

For the case of a circular orbit at the onset of orbital decay, I imagine a steady decrease in tangential velocity, and a steady increase in radial velocity, as the satellite spirals inward. How can the tangential velocity increase when the only force acting in the tangential direction is air resistance (unless gravity is somehow causing both velocity components to continually increase)?

To quote from Larry Niven, "Forward takes you out, out takes you back, back takes you in, and in takes you forward." This is not contradictory, it is just very counterintuitive.

Suppose we have two vehicles in the same circular orbit, with one just a bit behind the other. Now suppose the lead vehicle applies an instantaneous delta-v against its orbital velocity vector (i.e., toward the other vehicle). For a very short period of time, the lead vehicle will move toward the trailing vehicle. The vehicle also starts to drop below the trailing vehicle. Now orbital dynamics kicks in. The delta-v places the lead vehicle in a lower orbit (smaller semi-major axis, to be precise). From the vantage point of the trailing vehicle, the lead vehicle now begins to pull away. Now instead of a single delta-v, have the lead vehicle apply delta-v continuously but in very small amounts (i.e., how drag works). So long as the drag force is small, the lead vehicle will simply fall down and forward: it speeds up. If you make the drag force large enough, the vehicle will slow down. However, the drag has to be very large for this to happen.

 

[ee78]

To quote from Larry Niven, "Forward takes you out, out takes you back, back takes you in, and in takes you forward." This is not contradictory, it is just very counterintuitive.

Suppose we have two vehicles in the same circular orbit, with one just a bit behind the other. Now suppose the lead vehicle applies an instantaneous delta-v against its orbital velocity vector (i.e., toward the other vehicle). For a very short period of time, the lead vehicle will move toward the trailing vehicle. The vehicle also starts to drop below the trailing vehicle. Now orbital dynamics kicks in. The delta-v places the lead vehicle in a lower orbit (smaller semi-major axis, to be precise). From the vantage point of the trailing vehicle, the lead vehicle now begins to pull away. Now instead of a single delta-v, have the lead vehicle apply delta-v continuously but in very small amounts (i.e., how drag works). So long as the drag force is small, the lead vehicle will simply fall down and forward: it speeds up. If you make the drag force large enough, the vehicle will slow down. However, the drag has to be very large for this to happen.

Hmm, I think that I understand:

At the very beginning of the process, when the satellite is in, for the sake of simplicity, a circular orbit, air resistance briefly decelerates the satellite. As a result, it starts to accelerate radially inward and gain kinetic energy. Since gravity acts as a steering force, in that it prevents an object from flying off into space at a tangent to its orbit, this increase in kinetic energy is "shared" between the radial and tangential components (as opposed to the increase only going to the radial component), and thus, both tangential and radial velocity increase as the satellite spirals inward from its initial circular orbit. Since the air resistance is small, both components of velocity continue to increase, until the air resistance becomes large (analagous to skydiving before hitting terminal velocity).

Is this understanding of the phenomenon correct?