Understanding Hyperbolic & Parabolic Trajectories

[Maria76]

Hi,

I'm trying to wrap my head around the different kinds of trajectories. Firstly, what kind of objects have hyperbolic trajectories? And how do we differentiate between a parabolic trajectory and a hyperbolic one?

Thank you for any help.

Maria

 

[tiny-tim]

Hi Maria!

Firstly, what kind of objects have hyperbolic trajectories? And how do we differentiate between a parabolic trajectory and a hyperbolic one?

(we're talking about gravitational trajectories?)

If the planet or comet returns, it has an elliptic trajectory.

If it passes the star only once, and doesn't return, it has a parabolic or hyperbolic trajectory.

A parabolic trajectory has only one asymptotic direction … it both comes from and returns to that direction. It "reaches infinity" with zero speed and energy (and an object launched non-vertically with escape velocity goes into a parabolic trajectory).

A hyperbolic trajectory has two (non-parallel) asymptotes … it comes from one and returns to the other. It "reaches infinity" with positive speed and energy.

 

[spacester]

Hi,

I'm trying to wrap my head around the different kinds of trajectories. Firstly, what kind of objects have hyperbolic trajectories? And how do we differentiate between a parabolic trajectory and a hyperbolic one?

Thank you for any help.

Maria

When a spacecraft does a fly-by maneuver past a planet in order to gain velocity, it has a hyperbolic trajectory relative to the planet. It is going too fast to be captured by the planet, but close enough to have its course changed.

If you take that same spacecraft and aim for the same altitude above the planet's surface, but slow it down enough, it will get captured in orbit (its energy becomes 'bound' to the gravity field). This is of course an elliptical trajectory.

A parabolic trajectory is at the balance point between hyperbolic and elliptical. Using that same spacecraft , any faster than parabolic is hyperbolic; any slower is elliptical.

Hyperbolic orbits are rare in nature but do exist. When a Near Earth Object passes closely by Earth, it has a hyperbolic trajectory relative to Earth. Of course, it also is in an elliptical orbit relative to the sun. That elliptical orbit will be shifted because of the energy transfer between Earth's gravity field and the NEO during the brief time while the NEO is in the hyperbolic "orbit".

An asymptotic trajectory is unfamiliar to me and seems like more of a math professor problem than a real-world situation.