Why is the earth's orbit elliptical? That is, what force acts on the earth perpendicular to the force of the sun, that causes the tangential accelerations characteristic of an ellipse?
Since nobody has answered your question directly, I will. There is no tangential acceleration (at least, not in the simple two body problem of Newtonian mechanics). No tangential acceleration is needed.Why is the earth's orbit elliptical? That is, what force acts on the earth perpendicular to the force of the sun, that causes the tangential accelerations characteristic of an ellipse?
The acceleration is still directed against the position vector for an elliptical orbit (or a parabolic or hyperbolic orbit, for that matter). What makes you think there has to be a tangential acceleration?There is indeed a tangential acceleration. Acceleration is change in velocity. For a circular orbit, the change is perpendicular to the velocity, and parallel to the radius. For an elliptical orbit, the velocity changes in magnitude, and so the acceleration has a component perpendicular to the radius.
Where am I going wrong?
This is incorrect.Let me phrase my comments in another form.
Consider the two-body problem of the earth and sun, each replaced by point masses. Define let the coordinate system so that the orbit is in the x-y plane. At t=0, let the earth be at its closest distance, and moving in the y direction. Solve the equations of motion, using the initial conditions. The solution is a circular path.
The reality is that the orbit of the earth is an ellipse, with an eccentricity ?, which is a function of time. Since the fact of ?(t) is a fact that does not occur in the initial equations, there is no way to find a solution with ? ? 0.
I believe that the eccentricity is a result of the many-body problem, taking into account the large planets. There is no analytical solution to the Newtonian gravitational many-body problem. I do not have the tools to solve this problem. I wonder if any of you know about many-body calculations that result in the correct ?(t)?
It's all in the initial conditions.No, I do not understand how a two body problem can give non-circular orbits, for then the result is more than the input.
It is not possible to get more out of a set of equations than the stuff that goes into the equations and initial conditions.
Suppose the mass of the orbital body [itex]m[/itex] is miniscule compared to the mass of the central body [itex]M[/itex] (i.e., [itex]m<<<M[/itex]). The circular orbit velocity isI think I got it. Consider my setup: A mass m is orbiting in the xy plane. At time t=0, the mass is at (x,0). It has a velocity v in the y direction. This is the given. Question: What will the orbit be?
This is the specific energy of the orbit,Can one write the eccentricity as a function of velocity - velocity(circle), derived from the equation of motion?
Did you sleep through your junior-level classical mechanics class?My level: Ph.D. theoretical physics.
Write your equations in LaTex, as [noparse][tex] LaTeX [/tex][/noparse], or use vBcode, [noparse] subscript [/noparse].BTW, how do I write subscripts in these messages?