How can you tell what kind of orbit a body will have?

In summary, the orbit of a body around another, or rather of both around their centre of mass, can be determined to be circular, elliptical, hyperbolic, or parabolic based on the object's current velocity. The specific values and conditions for each type of orbit can be derived from the equations of conservation of energy and angular momentum. The eccentricity of the orbit is determined by the ratio of the kinetic and potential energies, and the sign of the total energy determines the shape of the orbit. The circle is a special case of an ellipse, and the velocity can be calculated using the equations provided.
  • #1
21joanna12
126
2
Is there a way of telling whether the orbit of a body around another, or rather of both around their centre of mass, will give the object in question a circular, elliptical, hyperbolic or parabolic orbit?

Thank you!
 
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  • #2
If you know the objects' current velocity, you can calculate the trajectory/orbit.
 
  • #3
On a circular orbit, the speed is, for one, always at a right angle to radius. For another, it has a specific value. For a test body in the field of a point primary, since the centrifugal acceleration is vˇ2/R and gravitational acceleration is GM/Rˇ2, it means that the speed has to be the specific value of square root of (GM/R).

  1. So the orbit is circular if the speed is exactly square root of (GM/R) and exactly at the right angle to radius.
  2. If the speed is exactly square root of (2GM/R) then the orbit is parabolic unless v is collinear with R, in which case it is a straight line (and goes to infinity if it is away from primary)
  3. If the speed is any value bigger than square root of (2GM/R) then the orbit is hyperbolic unless v is collinear with R, in which case it also is a straight line and goes to infinity if it is away from primary.
  4. If the speed is any value less than square root of (2GM/R) but over zero then the orbit is elliptical except in two special cases - in case it is collinear with R, in which case it is a segment of straight line that does not go to infinity if it is away from primary, and the other special case stated in point 1), of the speed being both exactly square root of (GM/R) as well as exactly right angle to radius
  5. If the speed is zero then the orbit is the radius.
 
  • #4
snorkack said:
On a circular orbit, the speed is, for one, always at a right angle to radius. For another, it has a specific value. For a test body in the field of a point primary, since the centrifugal acceleration is vˇ2/R and gravitational acceleration is GM/Rˇ2, it means that the speed has to be the specific value of square root of (GM/R).

  1. So the orbit is circular if the speed is exactly square root of (GM/R) and exactly at the right angle to radius.
  2. If the speed is exactly square root of (2GM/R) then the orbit is parabolic unless v is collinear with R, in which case it is a straight line (and goes to infinity if it is away from primary)
  3. If the speed is any value bigger than square root of (2GM/R) then the orbit is hyperbolic unless v is collinear with R, in which case it also is a straight line and goes to infinity if it is away from primary.
  4. If the speed is any value less than square root of (2GM/R) but over zero then the orbit is elliptical except in two special cases - in case it is collinear with R, in which case it is a segment of straight line that does not go to infinity if it is away from primary, and the other special case stated in point 1), of the speed being both exactly square root of (GM/R) as well as exactly right angle to radius
  5. If the speed is zero then the orbit is the radius.
Thank you so mch for this! I was wondering if you have a link to derivations of these results?
 
  • #5
Bump! Does anyone have a derivation for the results above?
 
  • #6
Write down the possible velocities like so:
##0<V_c<V_e<\infty##
which corresponds to:
an ellipse for ##V## between ##0## and ##V_e## with the special case of a circle at ##V_c##, a parabola for ##V_e## and a hyperbola for anything larger.

You wrote the conditions for ##V_c## yourself in your recent thread about binaries. Just assume M>>m and you'll get the equation snorkack wrote.
The case of parabola is the case of V being equal to escape velocity. It's easily derived from conservation of energy:
http://en.wikipedia.org/wiki/Escape_velocity

As for why such progression can be assumed:
Orbits are conical sections as shown here:
300px-Conic_Sections.svg.png

where the difference between them can be reduced purely to eccentricity.
As can be seen here:
http://en.wikipedia.org/wiki/Orbital_eccentricity#Definition
here:
http://en.wikipedia.org/wiki/Laplace–Runge–Lenz_vector#Derivation_of_the_Kepler_orbits
or here:
http://en.wikipedia.org/wiki/Kepler_problem#Solution_of_the_Kepler_problem
##e=\sqrt{1+\frac{2EL^2}{k^2m}}##
the eccentricity is a function of orbital energy and angular momentum. But since the latter is squared, the sign of total energy E determines whether eccentricity ever goes below 1.

When total energy is less than 0 (KE < PE) the result is eccentricity of less than 1 (ellipse), if total energy
equals 0 (KE=PE) the eccentricity is 1 (parabola), and if it's more than 0 (KE>PE) the result is a hyperbola.
The circle is here a special case of an ellipse.

For a given distance from the central body (i.e., equal PE) the sign of total E is dependent only on the value of KE (i.e., velocity).

edit: fixed the sign in the equation
 
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Related to How can you tell what kind of orbit a body will have?

1. How do you determine the shape of an orbit?

The shape of an orbit is determined by the eccentricity of the orbit. This is a measure of how much the orbit deviates from a perfect circle. A highly eccentric orbit will have a more elongated shape, while a low eccentricity orbit will be closer to a circle.

2. What factors influence the type of orbit a body will have?

The type of orbit a body will have is influenced by its velocity, the mass of the body it is orbiting, and the distance between the two bodies. These factors can affect the shape and size of the orbit, as well as the time it takes for the body to complete one orbit.

3. Can the type of orbit change over time?

Yes, the type of orbit a body has can change over time. This can be due to various factors such as gravitational interactions with other bodies, atmospheric drag, and changes in the body's mass or velocity. However, these changes are typically gradual and may take a long time to be noticeable.

4. How does the angle of inclination affect an orbit?

The angle of inclination refers to the tilt of an orbit relative to a reference plane. This can have a significant impact on the shape and orientation of an orbit. For example, an orbit with a high inclination will have a more tilted and elongated shape, while a lower inclination will result in a more circular orbit.

5. Is there a mathematical formula for determining the type of orbit a body will have?

Yes, there are several mathematical formulas and equations that can be used to determine the type of orbit a body will have. These include Kepler's laws of planetary motion, Newton's laws of motion, and various equations related to gravitational forces and orbital mechanics.

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