FrontierScience / 20daab54-df1b-4006-a544-7198daee7847 / Suppose we have an object with large mass `M`. An object with mass `m` moves within the gravitational field created by `M`. Assume `m≪M`. Assume the angular momentum of `m` relativ…

Suppose we have an object with large mass `M`. An object with mass `m` moves within the gravitational field created by `M`. Assume `m≪M`. Assume the angular momentum of `m` relative to `M` is `L`, and the total energy is `E`. Assume `E<0`. Assume the motion is in the xy-plane with `M` at the origin. When `m` is closest to `M`, it is on the x-axis, and at this time, the velocity of `m` is in the +y direction. Derive the equation of the curve formed when plotting the trajectory of `m` in velocity space (with `v_(x)=(dx)/(dt)` and `v_(y)=(dy)/(dt)` as the two coordinate axes). Give your answer in the form `v_(x)²+(v_(y)−a)²=b`, where `a and b are expressions in terms of` `G` (the universal gravitational constant), `M`, `m`, `L`, `E`, `v_(x)`, and `v_(y)`.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

Suppose we have an object with large mass `\( M \)`. An object with mass `\( m\)` moves within the gravitational field created by `\( M \)`. Assume `\( m \ll M \)`. Assume the angular momentum of `\( m \)` relative to `\( M \)` is `\( L \)`, and the total energy is `\( E \)`. Assume `\( E < 0 \)`. Assume the motion is in the xy-plane with `\( M \)` at the origin. When `\( m \)` is closest to `\( M \)`, it is on the x-axis, and at this time, the velocity of `\( m \)` is in the +y direction. Derive the equation of the curve formed when plotting the trajectory of `\( m \)` in velocity space (with `\( v_x = \frac{dx}{dt} \)` and `\( v_y = \frac{dy}{dt} \)` as the two coordinate axes). Give your answer in the form `\( v_x^2 + (v_y - a)^2 = b \)`, where `\( a \) and \( b \) are expressions in terms of` `\( G \)` (the universal gravitational constant), `\( M\)`, `\( m \)`, `\( L \)`, `\( E \)`, `\( v_x\)`, and `\( v_y \)`.

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.