course Phy 201 ic_class_09207Orbital Mechanics
Class 091207
Suggested approach to problems:
First list the names, symbols, units and values of the quantities you are given. This helps get your thinking into the context of the situation. Be sure you're using the right words, the right symbols, the right units. The better you do with this, the more clearly you are likely to think about the situation.
Unless it's obvious don't even think about what the question asks you to figure out. Think first about what you can do with the information you have, rather than confining your thinking to a specific goal. Ask yourself what you can figure out from the given information. Use sketches, diagrams, words, etc..
Once you know what you can do with the information, it's time to focus yourself on the goal. This is the time to start thinking about what the question wants you to figure out.
Orbital Mechanics
`q001. Answer the following:
What quantities are associated with a satellite in circular orbit around a planet?
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The quantities associated with a circular orbit are to following:
- planet mass M
- satellite mass m
- orbital radius r
- gravitational force F = G M m / r^2
- centripetal acceleration v^2 / r of the satellite, and centripetal force m v^2 / r exerted on the satellite
- orbital velocity v, which is v = sqrt(G M / r)
- orbital KE, which is 1/2 v v^2 = 1/2 G M m / r
- PE of the satellite relative to infinity, equal to - G M m / r
- orbital period T
- orbital circumference 2 pi r
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Given the mass, velocity and orbital radius of a satellite in a circular orbit, what other quantities can we find, and what specific reasoning process do we use?
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From mass and velocity we can determine the KE of the orbit, and hence the gravitational PE (which is double the magnitude of the KE, and negative).
From the orbital radius and velocity we can use v = sqrt( G M / r) to find the mass of the planet.
From velocity and orbital radius we can find centripetal acceleration, which with the mass allows us to find the centripetal force and hence the gravitational force (which is equal to the centripetal force).
From the radius we can find the circumference of the orbit, which we can combine with the velocity to get the orbital period.
Thus we can find all the quantities listed previously.
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If we give a quick impulse to a satellite in circular orbit, in the direction of its motion, what can we predict will happen to the shape of the orbit, and to the kinetic, potential and total energies?
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The total energy of the satellite will increase. It will return to the point at which the impulse was delivered. However after the impulse it will be moving too fast to remain in the circular orbit, since the centripetal acceleration v^2 / r which would now be required to keep in in a circular path is greater than the gravitational force (which at this point is still equal to the previous centripetal force).
As a result the satellite moves into an elliptical orbit. It moves further away from the Earth, losing KE as its PE increases. Velocity v decreases as distance r from Earth increases, until the satellite reaches the point where its velocity is again perpendicular to the radial vector. At this point (the apogee of the orbit) it reaches its maximum distance from the Earth. Its speed at this point is insufficient for centripetal force m v^2 / r to match the gravitational force, and it begins moving back toward the Earth. Its KE increases as its PE decreases; the satellite continues to speed up until it reaches the point of the original impulse, at which point its velocity is again perpendicular to the radial vector.
It will then repeat the same elliptical orbit, and in the ideal case will continue to do so.
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Suppose a satellite is in an elliptical orbit. What difference does it make to the shape of the new orbit if a given impulse is delivered at the perigee vs. at the apogee? What difference does it make to the total energy of the new orbit?
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At whichever point the impulse is delivered the satellite will return to this point.
If the impulse is delivered at the perigee the satellite, upon returning to this point, will again be at its perigee (but moving now with increased speed and greater total energy). With its increased total energy its PE at apogee will be greater, so the apogee will be further from the Earth than before. This will elongate the ellipse.
A small impulse delivered at the apogee will also increase the energy of the orbit. The apogee will remain the same, since the satellite returns to the point of the impulse. However the PE at the perigee will be increased, so the perigee will move further from the Earth. Thus the elongation of the ellipse will decrease.
A sufficient impulse could increase the PE, and therefore the distance from Earth, at the perigee until that distance is equal to the distance at the apogee, in which case the satellite would be in a circular orbit.
An even larger impulse would further increase the distance at the opposite side of the orbit, so that the apogee becomes the perigee and the ellipse becomes elongated in the opposite direction.
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Which has greater total energy, a circular orbit or an elliptical orbit which remains inside the circular orbit except at two 'extreme points' where it just touches the circular orbit?
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The center of the Earth is at one focus of the ellipse, which doesn't coincide with the center of the circle. So the ellipse can't touch the circle at its two 'extreme points'. Your instructor posed this question without thinking.
However if the apogee of an elliptical orbit coincides with a circular orbit, the elliptical orbit will then being moving closer to the Earth. This indicates that the velocity of the object in the elliptical orbit is less than that of the circular orbit. Since the PE would be the same at both points, the total energy of the elliptical orbit would be less than that of the circular orbit (assuming equal satellite masses).
If the perigee of an elliptical orbit coincides with the circular orbit, then since the elliptical orbit will then move further from the Earth its total energy must be greater than that of the circular orbit.
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