Hyperbolic, Parabolic or Elliptical Orbit?

[cj]

A comet is first seen at a distance of d AUs from the Sun and is traveling with a speed of q times the Earth's speed.

Apparently it can be shown that if q²·d is greater than, equal to, or less than 2, then the comet's orbit will be hyperbolic, parabolic or elliptical respectively.

Any idea how this can be shown??

I know that, in general, ε (eccentricity) is less than, equal to, or greater than 1 for an ellipse, parabola, and hyperbola respectively.

 

[BobG]

Yes, see post: https://www.physicsforums.com/showthread.php?t=40525

Just substitute the heliocentric gravitational constant for the geocentric gravitational constant.

The specific energy of object (energy per unit of mass) is just:

$$\frac{v^2}{2}-\frac{\mu_{sun}}{r}=\varepsilon$$
where v is velocity, r is position and $$\mu_{sun}=1.327124421\times 10^{11}km^3/sec^2$$

If the specific energy is less than 0, the object will orbit the Sun. If equal to 0, the object will follow a parabola. If greater than 1, the object will follow a hyperbola.

'e' is normally used to represent eccentricity (depends on the book you're using)

 

[GSXR750]

This is because the eccentricity is related to the shape of the orbit, with a circular orbit having an eccentricity of 0, an elliptical orbit having an eccentricity between 0 and 1, a parabolic orbit having an eccentricity of 1, and a hyperbolic orbit having an eccentricity greater than 1.

To show the relationship between q, d, and the eccentricity of the comet's orbit, we can use the Vis-viva equation, which relates the speed of an object in orbit to its distance from the central body:

v^2 = GM(2/r - 1/a)

Where v is the speed of the object, G is the gravitational constant, M is the mass of the central body (in this case, the Sun), r is the distance between the object and the central body, and a is the semi-major axis of the orbit.

Since we know that the comet's speed is q times the Earth's speed, we can write:

v = q·v_Earth

Plugging this into the Vis-viva equation and rearranging, we get:

q^2·v_Earth^2 = GM(2/r - 1/a)

We also know that the distance of the comet from the Sun is d AUs, which can be converted to meters using the conversion factor 1 AU = 1.496x10^11 meters. So we can write:

r = d·1.496x10^11 meters

Substituting this into the Vis-viva equation, we get:

q^2·v_Earth^2 = GM(2/(d·1.496x10^11) - 1/a)

Now, the semi-major axis of an orbit can be calculated using the formula:

a = r/(1 - ε^2)

Where ε is the eccentricity of the orbit. Substituting this into the previous equation, we get:

q^2·v_Earth^2 = GM(2/(d·1.496x10^11) - 1/(r/(1 - ε^2)))

Simplifying and rearranging, we get:

q^2·v_Earth^2 = GM(2 - (d·1.496x10^11)/(1 - ε^2))

This equation relates the comet's speed, distance from the Sun, and eccentricity. We can now analyze the three cases mentioned in the content: