In another thread recently I suggested that the earth is ever so slowly slowing down and that someday it would spiral into the sun. Of course the sun will be long dead by then but that’s not the point. Someone else suggested the earth, due to slowing down, suggested the opposite (the earth would move away from the sun).
I have to admit I’m puzzled. I understand that the smaller the radius of an orbit the faster you need to go to maintain that orbit. E.g. A satellite orbiting the earth 20 miles up needs to move faster than a satellite orbiting 50 miles up.
The perverse thing about it then is that our satellite orbiting at 20 miles up needs to slow down to achieve a higher orbit. On the flip side instinct tells me if you slow down you’ll fall back to earth.
It’s a lovely little paradox, isn’t it? If you’re confused by it, you’re in good company. The Gemini astronauts ran smack-dab into this problem during their first attempt at an orbital rendevous. The object was ahead of them, so they tried to speed up to catch it . . . but burning their rockets moved them into a higher, and therefore slower, orbit, so they fell further behind. After quite a bit of consternation, they realized that what they had to do was turn around and fire their rockets the other way, to fall into a lower, faster orbit, get a bit ahead of the target, then return to a higher, slower orbit and drop back toward it.
The way I keep it straight in my mind is by not thinking in terms of orbital speed, but orbital energy. The higher the orbit, the higher its energy. So to get to a higher orbit, you need to do a burn that intuition would tell you would make you go faster, but really it’s just increasing your orbital energy.
BTW, because the Earth rotates faster than it orbits, it doesn’t always keep one face toward the Sun. The Sun raises a tidal bulge on the Earth, and then pulls on it as the Earth rotates, so tides are working to slow the Earth’s rotation. Through a bit of complicated accounting, the end result is to take energy out of the Earth’s rotation and put it into the Earth’s orbit, pushing it into a higher orbit. This is an extremely slow process, but it’s faster than any other effects you might be thinking of like gas drag.
That is somewhat misleading, because you have perturbed the orbit by applying a force. But clearly the earth could not go slower in a lower orbit without spiralling into the sun, if it were perhaps losing energy to frction ofrsomething like that.
Suppose you start off in a circular orbit, with radius 1 AU (a fair approximation to the Earth’s orbit). If you decrease your orbital speed suddenly, you’ll go into an elliptical orbit, with apihelion distance (furthest distance from the Sun) of 1 AU. At apihelion, you’ll be going slower than you were originally… But as you fall closer to the Sun, you’ll lose potential energy and therefore gain kinetic energy. When you hit perihelion (closest approach to the Sun), your speed will actually be greater than it was in the original circular orbit. Now, suppose that when you’re at perihelion, you again suddenly decrease your speed by the right amount. Now, you’ll be in a lower circular orbit, with a speed greater than your original speed.
In other words, if V[sub]c1[/sub] is your speed in your original circular orbit, V[sub]e1[/sub] is the speed of your elliptical orbit at apihelion, V[sub]e2[/sub] is the speed of your elliptical orbit at perihelion, and V[sub]c2[/sub] is the speed of your new, lower circular orbit, then V[sub]e2[/sub] > V[sub]c2[/sub] > V[sub]c1[/sub] > V[sub]e1[/sub] .
But back to the original question, if you’ve got some continuous drag acting on your orbit, rather than sudden changes of speed, your orbit will gradually get lower and faster.