why planetary orbits necessarily be ellipses???

why planetary orbits necessarily be ellipses???


1 Answers

Ramesh V
70 Points
13 years ago

Short Answer: It's generally because it requires a precise orbital velocity in order for a planet's orbit to be perfectly circular. If a planet's orbital velocity is anything other than that exact number, its orbit will be somewhat elliptical.

Longer Answer: Let's apply some math to the problem. Suppose a planet is orbiting a star with 1 Solar mass, 1.99e30 kg. Suppose its orbit is perfectly circular with a radius of 1.50e11 meters. The sunward acceleration of the planet (centripetal acceleration) is given by the equation:
a = GM/r²
Where G is the universal gravitational constant, 6.67e-11 Nm²/kg², M is the mass of the star in kilograms, and r is the sun-planet distance in meters. For our hypothetical planet, the sunward acceleration would be constant due to the circular orbit.

a = 5.90e-3 m/s²
Knowing the centripetal acceleration allows us to figure out how fast the planet must be moving in order to maintain a circular orbit:
a = v²/r
5.90e-3 m/s² = v² / (1.50e11 m)
v² = 8.85e8 m²/s²
v = 2.97e4 m/s
So a planet orbiting a star of 1 solar mass at a constant distance of 150 million kilometers would have to keep a constant orbital velocity of 29.7 kilometers per second. If the planet were to slow down (because of the gravitational influence of another planet, e.g.) then it would no longer have the orbital velocity needed to maintain its distance from the star. The planet would start to move closer to the star. However, the planet wouldn't spiral inward indefinitely; instead, the star's gravity would cause it to accelerate ever faster as it got closer, increasing its orbital velocity and causing it to slingshot around and back out to approximately where it was to begin with. So if the planet's orbital velocity decreases, its orbit becomes elliptical.




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