Physical Significance of Eccentricity & Semi-Latus Rectum of Orbital Ellipse

In summary, the eccentricity and semi-latus rectum of an orbital ellipse are ways of quantifying the deviation from a circular orbit and the distance of minimum approach or maximum distance of the orbiting body. They are related to the time period of the orbit and can be expressed mathematically using the harmonic mean and simple harmonic oscillation of the gravitational potential. The true anomaly and orbital radius can also be determined using these values.
  • #1
Clive Redwood
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What are the physical significances of the eccentricity and of the semi-latus rectum of the orbital ellipse?
 
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  • #2
They're ways of quantifying how different the orbit is from a circular orbit. The physical significance comes from the fact that an elliptical orbit is physically different from a circular orbit (the distance from the center changes with time, the speed of the orbiting body changes with time).
 
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  • #3
Clive Redwood said:
What are the physical significances of the eccentricity and of the semi-latus rectum of the orbital ellipse?
if you look up the shape of planetary orbits the eccentricity significantly describe the nature of path and the eccentricity alongwith semi latus rectum are related to the distance of minimum approach or maximum distance for say an elliptical path and in turn it gets related to time period of the planet.
 
  • #4
Please consider the following:

Assuming a simple harmonic oscillation of the gravitational potential centered at -GM/l, and with extrema labeled 1 and 2, then:

a. the shifts in the potential are equal and opposite:

-(GM/l - GM/r1) = -(-(GM/l - GM/r2))

Dividing by GM reveals l as the harmonic mean of r1 and r2.

b. Dividing the equation above by GM/l , we get:

1/r1 -1 = 1 - 1/r2 = e

This is the magnitude of the fractional shifts of the gravitational potential. It is also the eccentricity.

c. The amplitude of the oscillation is eGM/l . So the potential at a distance r may be expressed as:

- GM/r = - (GM+eGMcosq)/l

were q is a state variable of the oscillation. This equation may be rewritten as:

r = l/(1 + ecosq)

The couple (q, r) alternates between the extrema (0, r1) and (π, r2). These are 'collinear' with the 'origin'. So assigning the quantity 2A to the 'length' between these points, we get:

2A = l/(1 + e) + l/(1 - e)

and l = A(1 - e2)

So we may express the orbital radius as:

r = A(1 - e2)/(1 + ecosq)

This is, in polar coordinates, the equation of an ellipse. For the orbital ellipse, q is the true anomaly. Also l is the semi-latus rectum and is shown here to be the orbital radius at the center of the simple harmonic oscillation of the gravitational potential.
 

1. What is the physical significance of eccentricity in the orbital ellipse?

The eccentricity of an orbital ellipse represents the shape of the ellipse, with 0 being a perfect circle and 1 being a parabola. This value is important in determining the path of an object in orbit and how close it gets to the object it is orbiting.

2. How does the semi-latus rectum of an orbital ellipse relate to the shape of the orbit?

The semi-latus rectum is a measure of the size of the orbit and is equal to half of the length of the ellipse's major axis. It is directly related to the eccentricity of the orbit, with a higher eccentricity resulting in a smaller semi-latus rectum and a more elongated orbit.

3. What is the significance of the semi-latus rectum in orbital mechanics?

The semi-latus rectum is an important parameter in calculating the orbital period and velocity of an object in orbit. It also helps determine the closest distance an object will reach to the central body it is orbiting, known as the periapsis.

4. How do changes in eccentricity and semi-latus rectum affect an object's orbit?

A change in eccentricity will alter the shape of the orbit, with higher eccentricity resulting in a more elliptical orbit and lower eccentricity resulting in a more circular orbit. A change in semi-latus rectum will affect the size of the orbit, with a larger semi-latus rectum resulting in a larger orbit and a smaller semi-latus rectum resulting in a smaller orbit.

5. Can the physical significance of eccentricity and semi-latus rectum be observed in real-life orbital systems?

Yes, the eccentricity and semi-latus rectum of an orbital ellipse can be observed in various real-life orbital systems, such as planets orbiting around the Sun or moons orbiting around planets. These parameters play a crucial role in determining the stability and behavior of these orbital systems.

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