Max Aphelion Distance of Kuiper Belt Comet w/216 yr Orbital Period

[bri7]

Homework Statement

Find maximum possible aphelion distance for a Kuiper Belt comet with an orbital period of 216 years?
What is the eccentricity of this orbit?
What prevents a periodic comet from following this precise orbit?

Homework Equations

perihelion = a(1-e)
aphelion = a(1+e)
2a = perihelion + aphelion
P^2=a^3

The Attempt at a Solution

P^2=a^3
(216)^2 = a^3
a = 36 AU (semi major axis)

From here, I'm not sure where to go because I don't know the eccentricity, perihelion or aphelion distance

 

[phyzguy]

You've calculated the semi-major axis (I think correctly), and you are asked to find the maximum possible aphelion. What does maximum possible aphelion mean in terms of the perihelion? Is there a minimum possible perihelion that the object can have and still be a Kuiper belt object?

 

[bri7]

You've calculated the semi-major axis (I think correctly), and you are asked to find the maximum possible aphelion. What does maximum possible aphelion mean in terms of the perihelion? Is there a minimum possible perihelion that the object can have and still be a Kuiper belt object?

Okay so since the aphelion is the point where the comet is furthest form the Sun, the max aphelion is 71.99 AU, and then I can calculate the eccentricity of the orbit

 

[phyzguy]

Okay so since the aphelion is the point where the comet is furthest form the Sun, the max aphelion is 71.99 AU, and then I can calculate the eccentricity of the orbit

How did you come up with the 71.99 AU number?

 

[bri7]

How did you come up with the 71.99 AU number?

The major axis is 72 au and the max distance that the comet can be from the Sun is some number very close to 72, such that the orbit goes very close to the Sun at perihelon and very far away at aphelion
71.99 was just something I picked

 

[phyzguy]

If it grazes the sun at perihelion, would it be considered a Kuiper belt object?

 

[bri7]

If it grazes the sun at perihelion, would it be considered a Kuiper belt object?

No, I suppose it wouldn't

 

[bri7]

Is the definition of a Kuiper Belt object that it lies between 40 and 50 AU from the Sun?

 

[Emi11]

Here are my thoughts:
The inner edge of the Kuiper Belt is 30 AU, outer edge is 50 AU. Hence, the closest the comet's orbit can get to the Sun is 30 AU. This is our perihelion, because in order to maintain that the semi major axis is 36, and we want the maximum possible aphelion, we calculate with the minimum possible perihelion.

So, we use the equation; semi major axis = (aphelion + perihelion)/2, giving us
36= (A+30)/2
A=42
The max possible aphelion is 42 AU.

 

[bri7]

Here are my thoughts:
The inner edge of the Kuiper Belt is 30 AU, outer edge is 50 AU. Hence, the closest the comet's orbit can get to the Sun is 30 AU. This is our perihelion, because in order to maintain that the semi major axis is 36, and we want the maximum possible aphelion, we calculate with the minimum possible perihelion.

So, we use the equation; semi major axis = (aphelion + perihelion)/2, giving us
36= (A+30)/2
A=42
The max possible aphelion is 42 AU.

Thanks, this makes sense :)
I am still wondering why this is not a valid orbit for a periodic comet

 

[Emi11]

Hmmm, well a short-period comet must have an orbit of 200 years, so perhaps the eccentricity should be lower in order for it to be a valid orbit for a short period comet? However, this doesn't tangibly prevent the comet from following this orbit so I am unsure.

 

[gneill]

Sorry for the late reply, but I just noticed this thread. I don't often browse here.

I am still wondering why this is not a valid orbit for a periodic comet

I think that's a clue that there's no restriction on the minimum perihelion distance for a comet. A comet is an object that passes through the inner solar system where it can be warmed by proximity to the Sun and rendered visible in our night sky. I believe that the designation of the comet as a "Kuiper Belt comet" refers to its place of origin, it being an object perturbed from the Kuiper belt. The aphelion of a periodic comet is a good indication of its origin distance.

So your initial line of though regarding the orbit intersecting the Sun looks valid to me. Look up "degenerate ellipse" or "radial trajectory".

Just because an orbit has a period it doesn't mean it will every have a chance to complete one