# Why does the energy for a circular orbit not lie in the pit of the effective potential?

According to the well-known graph of the effective potential, it is clear that the body enters orbit only starting with some energy and at this energy the orbit goes round. In an experiment in the program, I showed that there is an energy at which a body falls (9km/s), at a slightly higher energy (17km/s), the orbit becomes an ellipse and only then the orbit takes the form of a circle (30ks/s). Where does this intermediate elliptical orbit come from, if there should be a fall, then circular, then elliptical?

UPD

I'm not sure I understood your question correctly, but I think your discussion with the effective potential is invalid because effective potential is plotted by keeping the angular momentum constant. Whereas in your "experiment" you increase the angular momentum each time.

When you start from a very low orthoradial velocity, you are going to have a very eccentric trajectory with the apogee at the launch point. Then, when you increase the speed, the eccentricity decreases until it becomes zero: this is the circular trajectory. If you increase the speed again, the eccentricity starts to increase again but this time the launch point is the perigee. Beyond the escape velocity, the trajectory becomes hyperbolic.

To conserve angular momentum, you must decrease the radius as you increase the orthoradial velocity. The curve of the effective potential shows us what must happen. You start from a large radius and therefore from a small orthoradial velocity : the trajectory is very eccentric and the starting point is the apogee. Then, you decrease the radius and increase the orthoradial velocity: we arrive at a circular trajectory which effectively corresponds to a minimum of the energy for this angular momentum. Finally, as you continue to decrease the radius, the eccentricity increases again and your starting point is perigee.

Hope it can help !

• Thanks for the answer, I think you're right about the conservation of angular momentum, and that's why I conducted more experiments, but with the conservation of angular momentum. I attached a new picture where the red line shows all the positions in which the angular momentum is preserved. I moved the satellite along this line, but I didn't find a circular orbit anywhere. What am I doing wrong? Commented Dec 11, 2022 at 2:59
$$U_{eff} \left( \mathbf{r} \right) = \frac{L^{2}}{2 \mu r^{2}} + U \left( \mathbf{r} \right)$$
In fact, everything is described by the standard formula effective potential + kinetic energy of radial motion = total energy. The problem with my experiments is that too many parameters changed every time, bringing chaos to the experiment. First, the initial velocity must be perpendicular to the radius in order to remove the kinetic radial energy. Secondly, it is impossible to change the coordinate, because due to its change, both the shoulder at the moment of momentum and the denominators in the terms in the effective potential energy formula change immediately. In order to achieve a controlled experiment, it is necessary to change only the initial velocity module so that the angular momentum changes accordingly and, as a consequence, the effective potential energy graph changes. At a speed of $$0$$, you will get the usual graph of the potential of a gravitating body - $$1/r$$ and the body will simply fall. As you increase the speed, you will get a pit that will go from left to right and when you reach your initial movement coordinates, then the speed needs to be fixed. If you have given too little speed, the orbit will be with aphelion at the starting point. If there are too many then the perihelion will be at the starting point.