What is the relationship between escape velocity and orbital velocity?

[gokul.er137]

I am trying to understand the existence of orbits apart from the elliptical one. I have used the following line of thought.

Consider an object moving from infinity towards a planet. The object has kinetic energy alone at infinity. But it develops a potential energy as it comes closer to the planet. Thereby, its kinetic energy reduces. As it comes sufficiently closer to the planet, its kinetic energy reduces. At the radial distance wherein its speed is equal to the orbital speed of the planet, it starts to move around and orbit the planet. But if its speed throughout somehow manages to be larger than the orbital speed then it continues to move away from the planet, suffering only a light deflection.

Thus, I gather that it is imperative that the speed of the object at all points be larger than the orbital velocity. But then, the gravitational force only always tends to infinity. In other words, the gravitational force always affects the object no matter how far away it keeps on going. My analysis is that, if the change in velocity as the object moves closer and farther away from the planet is negligible, then the object movies in a hyperbola. But if the change is not negligible and not larger enough reduce the speed to the orbital speed, then I guess it moves in a parabola.

I am developing on the mathematics to follow this. But I would like to know if my analysis is right.

Thanks in advance.

 

[Drakkith]

As your object approaches a planet, wouldn't it get MORE kinetic energy from being pulled towards it?

 

[Janus]

I am trying to understand the existence of orbits apart from the elliptical one. I have used the following line of thought.

Consider an object moving from infinity towards a planet. The object has kinetic energy alone at infinity. But it develops a potential energy as it comes closer to the planet. Thereby, its kinetic energy reduces. As it comes sufficiently closer to the planet, its kinetic energy reduces. At the radial distance wherein its speed is equal to the orbital speed of the planet, it starts to move around and orbit the planet. But if its speed throughout somehow manages to be larger than the orbital speed then it continues to move away from the planet, suffering only a light deflection.

In a system where you assign zero potential energy at infinite distance, as you approach the planet ,he potential energy becomes negative and becomes more negative as the closer you get to the planet. Thus the potential energy goes down and the kinetic energy goes up.

An object with exactly zero total energy (kinetic+potential) follows a parabolic path.
An object with positive total energy follows a hyperbolic path.
An object with negative total energy follows either a elliptic or circular orbit.)

Since an object falling from infinity cannot have less than zero total energy, it could only enter into a elliptic orbit if it sheds some of its kinetic energy in some manner.

 

[gokul.er137]

In a system where you assign zero potential energy at infinite distance, as you approach the planet ,he potential energy becomes negative and becomes more negative as the closer you get to the planet. Thus the potential energy goes down and the kinetic energy goes up.

Thanks. I messed up the signs. As you say, If the potential is 0 at infinity as it moves closer and closer to the planet, the potential energy should become more negative. I should however have understood intuitively that kinetic energy increases as an object comes closer and closer to the planet. Thanks to Drakkith too. I will post doubts as they plague me.

 

[gokul.er137]

An object with exactly zero total energy (kinetic+potential) follows a parabolic path.
An object with positive total energy follows a hyperbolic path.
An object with negative total energy follows either a elliptic or circular orbit.)

Since an object falling from infinity cannot have less than zero total energy, it could only enter into a elliptic orbit if it sheds some of its kinetic energy in some manner.

I gather that when you say 0 total energy the Kinetic energy exactly accounts for the potential energy at that point. Which implies that the velocity is sqrt(2*G*M/r), the escape velocity for that point. Is there any proof for this?