Why does Mercury have an elliptical orbit?

[TimeRip496]

Why is mercury obit elliptical? Is it because of the curvature of spacetime caused by the presence of sun such that the mercury moves in that orbit otherwise known as geodesics? However what about the other planets like Earth that orbits in a circular orbit? Is is just that that is their geodesics and attempt to explain such orbits through the curvature of spacetime is impossible due to the our inability to percieve the 4 dimensional spacetime?

 

[Nugatory]

You may be confusing elliptical orbits and precession.
Elliptical orbits are the standard solution for planetary motion in good old classical Newtonian physics, and all the planets follow at least slightly elliptical orbits - that's what you get if you solve the equations of motion for a $1/r^2$ force.

General relativity says that the Newtonian $1/r^2$ force law isn't exactly correct, and if you use GR to calculate the orbits you'll find that they are elliptical as Newton and Kepler said centuries ago, but also that the ellipses precess very slightly. This precession is most noticeable for Mercury because it is closest to the sun, the forces are greater, and the deviation from the classical non-precessing ellipse is greater.

All the planets are following geodesic paths through spacetime according to GR.

 

[Mentz114]

Is is just that that is their geodesics and attempt to explain such orbits through the curvature of spacetime is impossible due to the our inability to percieve the 4 dimensional spacetime?

No. There is a view on some crackpot sites that GR cannot describe the elliptical precessing orbits of planets. This is not true, and it is in fact Newton's theory which cannot describe the above.

The exact of solution of the ellipse like orbits uses the Weierstrass P function $\wp$.

https://en.wikipedia.org/wiki/Weierstrass's_elliptic_functions

 

[Vanadium 50]

This is not true, and it is in fact Newton's theory which cannot describe the above.

Technically, Newton can describe about 92% of the perihelion advance. It's only that last 8% that needs GR.

all the planets follow at least slightly elliptical orbits

Back when Pluto was a planet, it had a larger eccentricity than Mercury. (I guess it still does!) The asteroids 2 Pallas and 3 Juno also have larger eccentricities.

 

[Mentz114]

Technically, Newton can describe about 92% of the perihelion advance. It's only that last 8% that needs GR.

True.
To be more specific, the Newtonian solutions are conic sections which do not include precession at all. Obviously introducing perturbations and applying Newtonian theory can account for most of the precession. GR explains the so called anomalous precession.