Why are the orbits of planets elliptical?

These three answers are the best explaination i found on quora:

Answer 1 by Author: Robert Frost, Guidance, Navigation, and Control Instructor for NASA manned spacecraft
Newton figured out that any body under the influence of an inverse square force (e.g. gravity) will travel along a conic section.  The conic sections are the circle, the ellipse, the parabola, and the hyperbola.
Newton determined that any body orbiting the Sun will do so in an orbit the shape of one of these conic sections, with the Sun at a focus.  Something like this:
These orbits differ by their eccentricity:
  • Circle: 0
  • Ellipse: 0<e<1
  • Parabola: 1
  • Hyperbola: >1
Now that we’ve established the options for the orbits of planets, let’s figure out why they orbit in elliptical orbits.
1) The Solar system is 4.6 billion years old.  Any planets that had parabolic or hyperbolic orbits would be long gone.
2) A circular orbit requires achieving an eccentricity of exactly zero.  That’s hard. 
3) An elliptical orbit can have an eccentricity anywhere between 0 and 1.  That’s easy.
Answer 2 by Author: Richard Muller, Prof. Physics UC Berkeley, author “Physics for Future Presidents”
Actually, they are not elliptical. According to general relativity, the orbits will not close on themselves; they are almost elliptical, but not quite. Indeed, even in Newtonian physics, they are not elliptical unless the sun is a perfect sphere and there are no other planets.
Elliptical orbits come about only in two cases:
  1. Force is exactly inverse square. In this case, the center of the force is at one focus of the ellipse. Planetary orbits are approximately ellipses to the extent that the force is classical (no general relativity), dominated by the sun, by the fact that the sun is spherical (a quadrupole moment would change the law), and that there are no other planets present.
  2. Force is exactly linear (as in a spring). In this case, the center of the force is at the center of the ellipse.
Perhaps your question is: why, for an inverse square law, are orbits elliptical? I know no easy way to see why this is true. I am familiar with the usual physics derivation, but the result of elliptical orbits seems to come out of that as a surprise; it is almost miraculous. Of course, inverse square is a simple law, and so you might expect a simple result; but many other central forces are simple (e.g. exponential, as we have for the nuclear force; or inverse distance) but don’t yield such a simple result.
That presumes, of course, that you consider the elliptical orbit to be “simple”. Most non-mathematicians/physicists wouldn’t necessarily think so. It is remarkable that Kepler discovered that they are (approximately) elliptical.
Answer 3 by Author: Sandesh Patkar, Associate Research Strategist (2017-present)
To know why orbits of planets are elliptical, first you need to know little about Einstein’s theory of general relativity. What it says is that space-time is a fabric, like a cloth. Heavy objects bend space the same way a bowling ball will bend a cloth or a towel when placed upon it. See this video to know better: 

Now, having seen the video I assume you can imagine the shape formed by the ball. Let the ball equal to sun. The curvature in space when extended, will form a cone. See the paint image for an idea (1).

When you cut the cone through different positions, you get a circle, a ellipse, a hyperbola, or a parabola. Depending upon the eccentricity.

(Eccentricity: In mathematics, the eccentricity, denoted e or , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular. In particular, The eccentricity of a circle is zero. Source: wikipedia More: Eccentricity (mathematics)) See the other image (2).

The orbits of planets are elliptical only because they are cutting the cone at an eccentricity less than 1, condition necessary for an ellipse to form. It all depends at what eccentricity the planet is cutting this cone formed by curvature of space. Hope this answers your question! 🙂

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(2)
Source: Quora

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