Precession of the perihelion of Mecury

[tdotson]

My problem involves the precession of the perihelion of Mecury
F sub g = - (GMm)/r^2 * (1 + a/r) where a << r
I proved in previous parts d^2r/dt^2 - r*(dtheta/dt)^2 =
-GM/r^2 * (1 + a/r) [eqn 1] and r* d^2theta/dt^2 + 2*dr/dt*dtheta/dt = 0
I also used u(theta) = 1/r(t) to turn eqn 1 to d^2u/dtheta^2 +
u(1-GMa/l^2) = GM/l^2 where l = L/m.
Where I'm stuck is showing the solution is u(theta) = u(sub 0) *
(1+ ecos(n(theta - theta (sub 0))) where e and theta (sub 0) are constants of integration and u(sub 0) and n are in terms of a, G, M and l. I've tried many times but can't get it to work out.

 

[HallsofIvy]

Your equation is d^2u/dtheta^2 + u(1-GMa/l^2) = GM/l^2 which we can write as d^u/dtheta^2+ Au= B by setting A= 1-GMa/l^2 and B= GM/l^2.

The general solution to the differential equation d^u/dtheta^2+ Au= 0 is u= C1 cos(sqrt(A)theta)+ C2 sin(sqrt(A)theta). The constant solution u= B/A satisfies d^u/dtheta^2+ Au= B since the second derivative is 0. Adding those,

u= C1 cos(sqrt(A)theta)+ C2 sin(sqrt(A)theta)+ B/A is the general solution to the differential equation. You should be able to write that in terms of your constants.

You may need to use the facts that
cos(theta- theta0)= cos(theta0) cos(theta)+ sin(theta0)sin(theta) and
sin(theta- theta0)= cos(theta0) sin(theta)- sin(theta0)cos(theta) to put it in the form you want.

 

[M98Ranger]

The precession of the perihelion of Mercury is a well-known phenomenon in astronomy that has puzzled scientists for centuries. It refers to the gradual shift in the orientation of Mercury's elliptical orbit around the Sun, which cannot be fully explained by classical Newtonian mechanics.

Your problem involves the application of Newton's laws of motion to derive the equation of motion for a planet orbiting the Sun. From your previous calculations, you have derived an equation that describes the motion of a planet in polar coordinates, taking into account the influence of the Sun's gravitational force and a small perturbation term a/r.

To solve this equation, you have introduced a new variable u(theta) = 1/r(t) and substituted it into the equation, which has transformed it into a differential equation in terms of u and theta. This is a common technique used in celestial mechanics to simplify the equations of motion.

However, you are now stuck in trying to show that the solution to this differential equation is given by u(theta) = u(sub 0) * (1+ ecos(n(theta - theta (sub 0)))) where e and theta (sub 0) are constants of integration and u(sub 0) and n are in terms of a, G, M and l.

To solve this problem, you will need to use the method of separation of variables, where you assume that the solution can be expressed as a product of two functions, one depending only on u and the other only on theta. By substituting this into the differential equation and equating the coefficients of the two functions, you can then solve for u and theta separately.

Once you have obtained the expressions for u and theta, you can then substitute them back into your original equation and solve for r(t), which will give you the equation of motion for the planet's orbit. From there, you can use Kepler's laws of planetary motion to determine the constants of integration e and theta (sub 0) and relate them to the physical parameters of the system, such as the semi-major axis a and the angular momentum l.

In conclusion, solving the problem of the precession of the perihelion of Mercury involves a combination of mathematical techniques and physical principles. It requires a thorough understanding of classical mechanics and celestial mechanics, as well as the ability to manipulate and solve differential equations. Keep working at it and don't hesitate to seek help from your peers or instructors if you are still stuck.