Gravity Off?

Question

Kepler's Second Law states that the imaginary line between any planet and the sun sweeps equal areas in equal times as it orbits the sun. If the gravitation between the sun and the planets were somehow switched off and the planets no longer followed elliptical paths, would Kepler's equal area in equal time law still hold true?

a) Yes, the Second Law would still hold, even with gravity off.

b) No, Kepler's laws relate to elliptical orbits produced by gravity. Turn off gravity and Kepler's Second Law is meaningless.

The answer is: a. Kepler's Law states that the imaginary line joining a planet to the sun sweeps out equal areas o space in equal time. This just means that the planet's angular momentum around the sun is unchanging (Remember BALL ON A STRING). Now if gravity is shut off, the planet shoots away along the line of its direction of travel at constant speed. If the distances between the areas of space along the line of travel are all equal then the planet must ravel between them in equal times.

Why are the areas of space along the line of travel all equal? Because all triangles have equal bases and a common altitude at s (the sun in this case).

Answer 1

Yes, At least with respect to the Sun's reference frame, Kepler's law would still hold.

Newton's First and Second Laws of motion state:

"An object will remain at rest, or continue to move at a constant velocity, unless an external net force acts upon it," and

"Net force on an object is equal to its rate change of momentum"

If the Sun's gravity switched off, there would be no (or comparatively minor) net forces on the planets - at least from the Sun's reference frame. So the planets would travel at a constant velocity with respect to the Sun, and since there is no net force, there would be no change in momentum and there would be no change in direction.

The planets are now traveling in a straight line at a constant velocity relative to the sun. Therefore, they will go a constant distance per unit of time (d=v/t), which would be the base of a triangle. Geometery provides that the area of a triangle is 1/2 x height x base. Since the base is a constant and the height is the perpendicular distance from the base line to the sun, the areas will be the same for all triangles. So Kepler's Second Law holds!!!!! (Your first answerer came to the same conclusion).

As an aside, I have picked up a few principles of General Relativity recently. And I've learned (or relearned would be a better statement) that the planet's elipitical orbits are geodesics which are considered to be the equivalent of straight lines on curved space. So, for both sets of geodesics, the planets would trace out paths of equal areas. And in their frames of reference they would be going 'straight.' Interesting.

Moreover, my recollection (it is slowly coming back) is that Kepler's second law works for all orbits including parabolas and hyperbolas. A straight line is just a hyperbola taken to its limit.

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Eh Linlyons,

You'll get it. It sometime justs takes time to absorb and get those old neurons firing. Planetary orbits go from circles, to ellipses, to parabolas to hyperbolas to straight lines. Kepler's law, surprizingly, works on all of them. See http://en.wikipedia.org/wiki/Orbital_eccentricity (Orbital eccentricity) and http://en.wikipedia.org/wiki/Eccentricity_%28mathematics%29 (Eccentricity). The straight line is the easiest of all to solve for Kepler's law.

Linlyons (again). The principle of geodesics is that they are the shortest path through space-time (that is without adding an outside force) which is the equivalent of a straight line. The body in motion 'feels' no force, i.e., it feels weightless. A satellite in orbit follows a geodesic and it feels weightless. A planet's orbit is a geodesic and it's shortest path. Except for tidal forces, the planet feels no effects of gravity. Applying geodesics was just an interesting thought I had, and I have limite knowledge in that area, but I'm learning. :-)

Answer 2

That's a very interesting question. Without gravity, the regions swept out will be triangles rather than ellipse segments, as the planets will travel in straight lines.

Let's call that triangle ABC. A is the line between the sun and the planet's initial position. B is between the sun and the planet's final position. C is the path of the planet.

C will be the same length every day (because Newton's second law still applies)

A and B will get longer, and the angle between them will become smaller.

Proving the the triangle has constant area is more geometry than I can summon up right now. See if you can do it from my starting position.

Answer 3

Clearly not. Planets would travel in straight line paths at constant speed with the speed and direction they had at the moment gravity switched off. So, as time goes on, the planet will get farther and farther from the sun; since the planet is traveling at a constant speed, one leg of the imaginary triangle formed will be the same, however, the other leg of the triangle will be related to the distance of the planet from the sun and this will be an ever increasing distance. Therefore, the area of this triangle will increase as the planet travels farther from the Sun.

Answer 4

No, it would be meaningless. The planets would immediately begin traveling a trait line angling away from the sun and the area of distance traveled between the Earth and the Sun along the line would increase in a given period of time.

Answer 5

a) Yes.

Well, not exactly Kepler's Law anymore if gravity is turned off.
The area sweeped by the planet will be a triangle.
Area of triangle is half base * height.
The height is the same, and the base is directly proportional to time, since the planet will have no acceleration. :)

Answer 6

The motion of the planets are unpredictable if the Gravity is off. Because the net force acting on the planet is not zero. However Kepler's law may have to be changed in htat case.

Answer 7

If gravity shuts off, there will be no planets, as the matter in them will no longer clump together, and will disperse. There will also be no sun, and nothing else to orbit, and no means by which to orbit.

Answer 8

Kepler’s laws were obtained from and empirical situation… even so the elliptical orbits are caused by the gravitational force, if you turn it off the closed trajectory would be lost so 2 kepler law would be meaningless R//b)

Answer 9

a

Answer 10

Dr. H.

Why are you such a prick?