Understanding Hyperbolic Orbits: The Relationship Between Foci and Trajectory

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Homework Help Overview

The discussion revolves around hyperbolic orbits, specifically the placement of the orbited body and the trajectory of an object, such as a comet, in relation to the foci of the hyperbola. Participants are exploring the mathematical relationships that define these trajectories.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the correct placement of the orbited body and the implications for the trajectory equation. There is discussion about the validity of different forms of the hyperbolic orbit equation and how the definitions of variables affect the outcomes.

Discussion Status

The conversation is active, with participants providing feedback on each other's interpretations. Some guidance has been offered regarding the definitions of variables and their implications for the equations used, but no consensus has been reached on the best approach.

Contextual Notes

There is an ongoing examination of how to define the variables r and P in relation to the foci, which may affect the interpretation of the equations. Participants are considering different scenarios based on the placement of the gravitational source.

WWCY
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Homework Statement


Screen Shot 2017-11-26 at 3.46.21 AM.png


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

 

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WWCY said:

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.
 
haruspex said:
Looks right.

Hi, thanks for clearing that up!

Edit: Is r-P = 2a also a valid hyperbolic orbit?
 
WWCY said:
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.
 
haruspex said:
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
 
WWCY said:
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.
 
haruspex said:
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!
 

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