# Mercury's precession

I read in an article about Mercury's precession that Newton's law of gravitation predicts such precession of planets ;but fails to caluclate the precession of Mercury.But most of popular science books or other articles on the internet suggest that Newton predicts identical ellipses whereas the real orbital shape is like a rose petal. To conclude, Newton's also thought that the orbits should be like rose petal because of the perihelion advancement due to precession .Right ?

Edit 1

• Just a note to say that Newton himself didn't know about the precession of Mercury, it was found out much later. – Mozibur Ullah Dec 6 '17 at 9:02

If two bodies orbit each other and the mass of one of them is much larger than the mass of the other body, the equations of motion can be solved exactly. In polar coordinates the solution is given by: $$r(\theta)=\frac{l}{1+e\cos{\theta}}$$ This is the equation of an ellipse. As the OP correctly mentions, the actual orbit of Mercury looks more like "rose petals". This is because the actual period of the orbit is slightly less than the standard $2\pi$, so after every orbit the ellipse starts again slightly earlier, resulting in these "rose petals". This effect is known as the "perihelion precession".

What causes this effect? The main contribution is purely Newtonian, and is due to the attraction of the other planets. To first-order perturbation theory one can compute the correction to the orbit, yielding the precession of the perihelion. So it is correct to say Newtonian gravitation predicts "rose petals", at least qualitatively. In quantitative terms, the prediction for how much the perihelion should shift every year is off by a small however noticeable amount. This can only be explained by corrections due to General Relativity.

For the Newtonian contribution to Mercury's perihelion precession, a very simple method to compute it is presented in the paper Nonrelativistic contribution to Mercury’s perihelion precession by Price & Rush.

However it should be mentioned that Newton never did the calculations himself, rather these consequences of Newton's laws were explored in the 19th century.

• i wanted this answer. I can not get my hands on that paper on Mercury's perihelion . Thanks . – user1157 Dec 5 '17 at 22:02
• @user1157 Thank you! :) Nowadays there are surely many ways to get your hands on a scientific paper! Just search the title on Google – John Donne Dec 5 '17 at 22:06
• relativitycalculator.com/pdfs/mercury_perihelion_advance/… . This was the article. – user1157 Dec 16 '17 at 11:37

Newton's law of gravitation predicts a perfect ellipse for the orbit of a planet orbiting a star, only in the idealized case that there a just those two bodies (the planet and the star). However, in reality there will be many bodies orbiting the star, each with its own gravitational field that slightly perturbs the orbits of the other bodies, causing the orbits to deviate from perfect ellipses and slightly precess.

• So , Newton did predict the rosetta like orbits? – user1157 Dec 5 '17 at 21:46
• His laws certainly do. Whether he ever did the calculation, I do not know. – mmeent Dec 5 '17 at 21:51