# How does velocity affects the shape of orbit

by goleafsgo113
Tags: affects, orbit, shape, velocity
 Sci Advisor HW Helper P: 2,284 Another way: $$\epsilon=\frac{v^2}{2}-\frac{\mu}{r}$$ $$\epsilon$$ is the specific energy per unit of mass. It determines the size of your orbit. It always winds up being negative. The closer to zero, the greater the specific energy (In fact, if it reaches zero, you have a parabolic escape orbit - greater than zero, a hyperbolic escape orbit). $$\mu$$is the geocentric gravitational constant (or the gravitational constant for the object you're orbiting). $$v$$ is velocity. $$r$$ is the radius of your current position. If the velocity decreases, the size of the orbit decreases. Your current position hasn't changed (assuming you instantaneously changed the velocity). Due to conservation of energy, you have to return to the same point that you made your maneuver at. If you started with a circular orbit, that means your new point is apogee. Any change in velocity made at perigee, apogee, or at any point in a circular orbit affects the point 180 degrees opposite the most, but does not affect the point you made the maneuver at (you have to return to the same point you made your maneuver at). The formula for determing the semi-major axis of your orbit is: $$a=-\frac{\mu}{2\epsilon}$$ Notice that all terms on the right are constants, except for $$\epsilon$$ The closer $$\epsilon$$ gets to zero, the bigger the orbit. The radius of apogee is found by: $$r_a=a(1+e)$$ with $$e$$ being your eccentricity. Wherever you were at in the circular orbit when you decreased the velocity is the apogee point. You have to pass through the maneuver point again. Decreasing the velocity decreased the size of the semi-major axis (a). So e has to increase in order to keep the radius of apogee constant.