Understanding Hyperbolic Orbits: The Relationship Between Foci and Trajectory

[WWCY]

Homework Statement

I'm rather confused as to where the orbited body is placed, and where the orbiting trajectory lies in this figure.

Is it right for me to say that if I placed say, the Earth on the left hand focus, a comet with a hyperbolic trajectory will travel a path defined by the equation $r - P = -2a$?

Thanks in advance.

Homework Equations

The Attempt at a Solution

 

[haruspex]

Homework Statement

View attachment 215615

I'm rather confused as to where the orbited body is placed, and where the orbiting trajectory lies in this figure.

Is it right for me to say that if I placed say, the Earth on the left hand focus, a comet with a hyperbolic trajectory will travel a path defined by the equation $r - P = -2a$?

Thanks in advance.

Homework Equations

The Attempt at a Solution

Looks right.

 

[WWCY]

Looks right.

Hi, thanks for clearing that up!

Edit: Is r-P = 2a also a valid hyperbolic orbit?

 

[haruspex]

Hi, thanks for clearing that up!

Edit: Is r-P = 2a also a valid hyperbolic orbit?

Not if the sun is at the negative focus.

 

[WWCY]

Not if the sun is at the negative focus.

But if I do want to use r - P = 2a, and place my planet on the right hand focus, wouldn't I need to redefine r to originate from the right focus and P from the left? This gives r - P = -2a again.

Thank you

 

[haruspex]

But if I do want to use r - P = 2a, and place my planet on the right hand focus, wouldn't I need to redefine r to originate from the right focus and P from the left? This gives r - P = -2a again.

Thank you

It depends how you are defining r and P.
If you define r as distance from the left focus and P that from the right focus then for the gravitational source at the left focus it is r-P=-2a; for the gravitational source at the right focus it's r-P=2a.
But if you define r as distance from gravitational source and P as distance from the other focus then it is r-P=-2a whichever side the source is.

 

[WWCY]

It depends how you are defining r and P.
If you define r as distance from the left focus and P that from the right focus then for the gravitational source at the left focus it is r-P=-2a; for the gravitational source at the right focus it's r-P=2a.
But if you define r as distance from gravitational source and P as distance from the other focus then it is r-P=-2a whichever side the source is.

Thanks!