# Calculate Aphelion Distance & Velocity of Comet at Perihelion

• Unto
In summary, the comet has a perihelion distance of 0.6Au and a period of 341 years. To calculate its Aphelion distance, we can use the formula T^2 = a^3 and find a value of 48.8AU for the semi-major axis. To determine the velocity at perihelion, we can use the equation u = GM and substitute 1.5E11m for GM to convert to the MKS system. The final velocity is 54km. To calculate the time and speed required for a spacecraft to rendezvous with the comet at its perihelion position, we can use Kepler's relationship T^2 = a^3 and vis viva
Unto
A comet is observed to have a perihelion distance of 0.6Au and a period of 341 years. Calculate it's Aphelion distance. What is the velocity of the comet at Perihelion?

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The first thing I did was remind myself of some important terminology:

Perihelion - closest distance to the sun/star
Aphelion - furthest distance from the sun/star.

What I'm confused is whether Aphelion is the same as the Major Semi-axis or different.. Because I've done allsorts of calculations, and I keep getting a relativistic speed which is just dumb. So my values for Aphelion are all wrong.

How would I calculate Aphelion?

I can directly determine the semi-major axis by using $$T^2 = a^3$$ and that gives me a value of 48.8AU as the semi major axis. Do I add this to the Perihelion and say that the sum is the furthest distance away from the sun? Or am I missing a crucial piece here?

I even drew a diagram, I have no way of finding the eccentricity of the orbit without Aphelion :-(.

Help please. And lastly for working out the velocity at perihelion, what is 'r' in the equation:

Assuming that 'a' is the semi major axis.

K I did it again and this time used a diagram, I have Aphelion currently as 97AU..

K I got a final velocity of 54km..

I used:

u = GM

But for some reason you have to put 1.5E11m under the the GM (I don't know why but I've done it before and it resolves things). I think I put in wrong values for a and r though. I know a is semi major axis, but what would r be?

Btw, how would I use 'the orbit sweeps equal areas in equal times' to solve this question if possible?

Unto said:
K I got a final velocity of 54km..

I used:

u = GM

But for some reason you have to put 1.5E11m under the the GM (I don't know why but I've done it before and it resolves things).
It's because GM is measured in the MKS system and the distances are given in AU. 1.5e11m is 1 AU, So it gets everything into the same units.
I think I put in wrong values for a and r though. I know a is semi major axis, but what would r be?

Yeah I thought it would be something along those lines since 1.5E11m is 1AU.

Thank you for your help, I have 1 last question.

A ship from Earth is sent via Hohmann transfer orbit to rendevous with the comet in it's perihelion position. How long would the spacecraft take to reach the comet and at what speed would it be traveling at when it arrives? Assume the comet and Earth move in the same direction and on the same plane.

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I drew a diagram and subsequently became confused. It looks as if the craft would have to slingshot around the Sun to even attempt to get into the orbit of the comet. The question doesn't specify at which point in the orbit of the earth/comet is the craft shot...

The diagram I drew is merely an assumption (a pretty good one I think), but now I don't know what to do. How would I calculate the time taken? And the speed at that point?

Ok I've just worked out how I would find the time period and the speed (I would simply use vis viva again), but the diagram is confusing me. The comet has a Perihelion of less than 1AU, so how would I draw this diagram to make sure I'm doing the correct calculations?

K I drew a shoddy diagram. If the Hohmann transfer orbit is eliptical (obviously), then it will have a perihelion of 0.6AU (which is the comets perihelion) and then an Aphelion of 1AU (since it was shot from Earth and that is the origin of the orbit).

It's major axis would therefore be 1.6, half of that is a semi-major axis of 0.8AU.

Now that this is an elliptical orbit, we can use Keplers relationship T^2 = a^3 to find the time for 1 cycle in years, half of which should be when the craft rendevous with the comet which it what I want. I get a time period of 0.357 years... WTF?

I then used vis viva and got a velocity of 40kms-1

Something has gone wrong here, and I blame it on my diagram...

Also, if this is correct, then my craft is going slower than the comet at rendezvous and I need to give it some energy. How would I calculate the energy needed if the craft is 500kg?

Thanks for any help, I will get a good mark in this exam.

Last edited:
Unto said:
K I drew a shoddy diagram. If the Hohmann transfer orbit is eliptical (obviously), then it will have a perihelion of 0.6AU (which is the comets perihelion) and then an Aphelion of 1AU (since it was shot from Earth and that is the origin of the orbit).

It's major axis would therefore be 1.6, half of that is a semi-major axis of 0.8AU.

Now that this is an elliptical orbit, we can use Keplers relationship T^2 = a^3 to find the time for 1 cycle in years, half of which should be when the craft rendevous with the comet which it what I want. I get a time period of 0.357 years... WTF?
Looks good to me.
I then used vis viva and got a velocity of 40kms-1

Something has gone wrong here, and I blame it on my diagram...
Both numbers are right in there as far as being correct.
Also, if this is correct, then my craft is going slower than the comet at rendezvous and I need to give it some energy. How would I calculate the energy needed if the craft is 500kg?
Straight forward. What is the difference in velocity between comet and craft when they meet? How much kinetic energy does that equate to for a 500 kg mass?
Thanks for any help, I will get a good mark in this exam.

You are a star ^^

I just use the kinematic energy equation with the relative velocites and solve for k.e.

Many Thanks! ^^

## What is the formula for calculating aphelion distance and velocity of a comet at perihelion?

The formula for calculating aphelion distance is: D = a(1+e), where D is the aphelion distance, a is the length of the semi-major axis, and e is the eccentricity of the comet's orbit. The formula for calculating aphelion velocity is: V = √(G(M+m)(2/r-1/a)), where V is the aphelion velocity, G is the gravitational constant, M is the mass of the sun, m is the mass of the comet, r is the distance between the comet and the sun at aphelion, and a is the length of the semi-major axis.

## What is the significance of calculating aphelion distance and velocity of a comet at perihelion?

Calculating aphelion distance and velocity allows scientists to better understand the behavior and characteristics of comets. It can also provide insights into the formation and evolution of the solar system.

## How do scientists determine the values for the variables in the formula?

Scientists use various methods, such as observations and measurements, to determine the values for the variables in the formula. For example, the semi-major axis and eccentricity can be calculated from the orbital elements of the comet, while the mass of the comet can be estimated from its size and composition.

## Can the aphelion distance and velocity of a comet change over time?

Yes, the aphelion distance and velocity of a comet can change over time due to various factors such as gravitational interactions with other celestial bodies and outgassing of the comet's surface. These changes can be observed and studied by scientists.

## How accurate are the calculations of aphelion distance and velocity?

The accuracy of the calculations depends on the accuracy of the input data and the assumptions made in the formula. In general, the calculations can provide a good estimate of the aphelion distance and velocity of a comet, but they may not be exact due to the unpredictable nature of comets.

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