Yes, here it is, your worst nightmare (bwah ha ha!) If a small object is launched directly downwards from an orbiting satellite, what kind of path do observers on the satellite see the object follow? Does it just accelerate directly away? (This is based on a 3rd year physics exam question) ...ok does the object appear to move towards the satellite , at a later time?
So they just give it a shove towards the earth? Well it will just move off at constant velocity at first. Say the satellite was in a circular orbital. The extra kick to the object will knock it into some kind of slightly more elliptical orbit. If it is a big enough kick the orbit might become elliptical enough to graze the atmosphere so the object can lose some energy (speed) and perhaps eventually spiral to the ground, but this certainly doesn't have to be the end result. But ok, you are wondering what someone looking out of the bottom of the satellite sees? I guess they just see the object drift away and eventually come back again, etc, although I'm not sure whether the periods of the orbits will remain synchronised. The object might orbit slightly faster since it gains some energy from the kick, and the satellite loses a little bit (although you said the object is small so I guess we neglect that). Oooh, actually, the object will probably even come back to a height greater the original sattelite since it's orbit has become elliptical, and it had a bit of extra velocity at the height of the original sattelite, so it will need to actually climb out a little higher to reach the point where it has zero vertical velocity. Hmm yes I think this is what will happen.
Yes. Apparently one of the possible outcomes is that observers on the satellite see the object travel in a vertical loop and return to its start point, the launchpad, but only if the launch is horizontal (to the satellite's velocity vector), and from a circular orbit.
If you want the object to return to the satellite you'd have to fire it backwards and down. The speed component of its velocity has to be equal to that of the satellite itself. For instance, if the orbital velocity of the satellite is 7300 m/s, then you could fire it down at 5161.88 m/s and backwards at 2138.12 m/s. From the Earth frame this gives the object a horizontal speed of 5161.88 and a vertical speed of 5161.88 m/s for a total velocity of 7300 m/s. Here's why: The total energy of a satellite is given by \(E = \frac{mv^2}{2}- \frac{GMm}{r}\) where r is the radius vector of the satellite at any given point of its orbit. It can also be found by \(E = -\frac{GMm}{2a}\) where a is the semi-major axis of the orbit. thus \(\frac{mv^2}{2}- \frac{GMm}{r}= \frac{GMm}{2a}\) \(\frac{v^2}{2}- \frac{GM}{r}= \frac{GM}{2a}\) Which means that for any object passing through a point of radial vector r, and in orbit around a given body, the semi-major axis of its orbit depends on v. Since the period of the orbit is related to 'a' by \(T = 2 \pi \sqrt{\frac{a}{GM}}\) In order for two bodies in orbit, and passing through the same point, to have the same orbital period is for them to have equal magnitudes of velocity. (The direction of their velocities can be different)
1. The satellite has a circular orbit. 2. The additional velocity of the ejected test mass means it will have a slightly elliptical orbit. 3. The test mass and the satellite are in relative motion, the test mass has a less stable orbit than the satellite. During orbits: "..the body comes to the common initial position almost simultaneously with the station, approaching it from above."
That depends on the magnitude of the downward velocity and your definition of "almost simultaneously". If you fire it downward relative to the satellite at 1m/s its period will just that much longer than the satellite's for them to miss each other by 1m on the first orbit. This distance will increase by 1 meter for every successive orbit. At firing downward at 10 m/s causes them to miss by 115 m after the first orbit. 20 m/s makes them miss by 461 m, and 30 m/s makes them miss by 1 km.
The given equations of motion, which describe the radial vector components, are: \( \ddot x \;= \;3\Omega^2 x \;+ \; 2\Omega \dot y\) \( \ddot y \;= \;-2\Omega \dot x\) \( \ddot z \;= \;- \Omega^2 z\) "Show that the ejected body moves sinusoidally relative to the satellite along the x-axis, where the satellite is the center of motion, and has a period \( T\; = \;2 \pi / \Omega \)." How many forces are acting on the test mass (or the passive object) after launch? There's 'active' gravity from M, the earth's mass, then centrifugal and Coriolis forces (that's three), are there any others? And last but not least is the restriction that the test mass have a small velocity, relative to satellite S, which will be traveling at say 7 km/s, the test mass has 'extra' velocity of at most a few tens of m/s.
First, a minor correction to post #4 by Janus58 Correct. Wrong sign! Think of it this way: Bounded orbits have negative total mechanical energy. Using the correct sign leads to the vis-viva equation, \(v^2 = GM\left(\frac 2 r - \frac 1 a\right)\) Now for post #6. It is usually a good idea to define your terms when you introduce equations, noodler. Do you even know what your x, y, and z represent? These are a simplified form of Clohesy-Wiltshire equations, aka the CW equations, aka Hill's equations. These are linearized equations of motion. In other words, they are not exact. Except for his sign error, Janus is correct.
If that's the case, the object will always lag the satellite, and never return to it or move ahead of it, or move to a higher altitude, then? Can you explain what is meant by the following: ".. the motion of the test mass along its elliptical orbit is slightly non-uniform: in accordance with Kepler's second law, it will approach a point of intersection which is opposite the launch point a little earlier than the satellite. As a result, the test mass is in front of the satellite at exactly half the orbital period." Note: the vertical loop observed from the satellite isn't closed, the test mass "almost" returns
At a rough guess, I would say x y and z correspond to 3 spatial directions. After one orbit, how close does the object come to returning to the start point, and how far away is the satellite at this point? What percentage of the greatest distance of separation between the two orbiting bodies is this? Janus58 is correct about the time-difference. But the question is about the path taken by the vertically launched object as seen from the satellite, determining orbital periods is part of the answer.
Since I have the question and the answer I'm quite sure that the following isn't an answer, and would probably not get any marks at all (or maybe 1 mark for trying): The names of the people who 'discovered' the equations isn't the question, nor is the question about how exact they are. kurros has in fact, provided the closest answer, minus any details, Janus58 has some details but hasn't answered the original question. DH hasn't really suggested anything, but we now know what to call the equations, instead of just boring old "equations of motion".
What exact path the object takes as seen from the satellite depends on the initial conditions. If the velocity boost to the object is fairly small, then it will at first move inward and forward with respect to the satellite. accelerating at first. Once it reaches perigee, it will begin to slow its forward movement with respect to the satellite (though still moving away) and begin to climb back up vertically. It will eventually reach a point where it will be moving straight up with respect to the satellite. It will then start to move backwards towards the satellite again while still climbing. When it reaches its apogee, it will start falling back down at will begin to slow its backwards speed. It will pass over head of the satellite, and eventually pass downward behind the satellite. With each successive orbit this path will be offset. Eventually, the satellite will no longer be inside the loop made by the object.
Right. One of the initial conditions is that the test mass have only a slightly larger velocity than the station. The other important condition is that the station has a circular orbit, and so is a stable 'center' w.r.t the test mass. How did you think this out, or did you do some back of the envelope stuff? (The answer is that observers on the station see what you describe - the test mass moving in a vertical elliptical path away from, then towards them) Do you think this is a "good" problem, or is it fairly run of the mill, just an exercise in lateral thinking and manipulating formulas? It also asks about the path of an object launched parallel to the direction of the satellite's motion. In this case the divergence is much larger. So methinks this is really about orbital divergence, given different conditions, and about investigating how the gravitational field does work on a test mass -- i.e. free fall, inert masses, etc.
I just more or less visualized it in my head from some orbital mechanics basics. I knew that the test object would gain energy, and thus it's average orbital distance would be higher and its period longer. I knew that after one orbit, it would cross the satellites orbit at the same speed and angle as it had when launched. I knew that the orbital speed at perigee is would be greater than that of an circular orbit at that same distance, and thus greater than the satellite's. For similar reasons, I knew that the apogee speed would be less. Thus the test object would pull ahead of the satellite while below it, start to lose that advantage when above it, and had to cross the satellite's orbit behind it. Well, you don't really need to use the formulas to solve this, as no numerical answer is needed. It's more of a test of knowledge of the general "rules" of Orbital mechanics.
The only thing I can think of now is, if the observers on the satellite didn't know they were in an orbit, would the motion of such test bodies, say a number launched in arbitrarily chosen direction, tell them anything about g 'forces'? What about orbiting around a black hole?
So, gratuitously, we advance on the main prize: "proof" that with a set of small test masses an astronaut can determine if they are orbiting a large massive body. Mind you, folks, sciforums is about as gratuitous as it gets, so don't mind me especially, will ya? I believe 'someone' here once disputed the above assertion about orbital divergence and being able to use it to detect the presence of a large mass. Please note that, placing a satellite in a circular orbit requires certain physical solutions to equations, but once in such an orbit there is no divergence from it except for local deformations in the field strength. In other words if it was orbiting a completely smooth object, it would be impossible to know this if there was no external (visual) frame of reference, like distant stars. If you had no way to see them, that would leave orbital divergence, with test objects, to determine orbital motion and period. Ha ha.
