4 Astronomy Problems: ASSISTANCE NEEDED PLEASE

[astronomystudent]

1. If a comet has an eccentricity of 0.98 and a semimajor axis of 12 AU, what are the apehlion and perihelion distances? What is the period? How does the velocity at perihelion compare to the velocity at aphelion?

In this problem, I used the perihelion and aphelion equations to solve for them, where P=(1-e)a and A=(1=e)a. Perihelion = 0.24 and Aphelion = 23.76. I don't know how to calcuate the velocity, and am confused by that. And since I have the eccentricity and semimajor axis, do I use the equation P(years) = R(A.U.)^3/2?

2. If this comet appeared to have a brightness of one at its nearest point to Earth, how bright would it appear at its maximum distance? Assume incorrectly that is does not change in luminosity as it approaches the Sun and use the inverse square law.

I know to use the inverse square law on this one, but how can I relate that to brightness and distances?

3. If an asteroid is in a 5:2 resonance with Jupiter, what is its semimajor axis?

The next two problems are the same and I don't have any equations for resonance but I will keep looking.

4. If an asteroid is in a 2:5 resonance with mars, what is its semimajor axis?

 

[tony873004]

1. Your answers for app and per are correct. Notice how apphelion and perihelon values are just (sma x ecc) +/- sma. That's basically what your formula says too. For velocities, you could just say that velocity at perihelion is faster than velocity at aphelion, which is correct for any orbit. It doesn't actually tell you to compute it, just to compare it. Off the top of my head, I forget the formula. See if you can find it.

2. This question doesn't really make sense. What is it's nearest point to Earth? All we're given is its solar elements. This comet could have high inclination, meaning that it's nowhere near Earth's orbit as it passes the 1AU point in its orbit. It's closest point to Earth could be a brush with our outer atmosphere (~100km altitude, 6470 km distance). It's furthest point from Earth would be roughly its furthest point from the Sun, + 1AU.

3. What is this asteroid's period? You must first know Jupiter's peroid. How does semi-major axis relate to period? (Kepler's 3rd Law).

4. Same as #3.

 

[SpaceTiger]

2. This question doesn't really make sense. What is it's nearest point to Earth?

Good point. If astronomystudent's teacher is thinking like an astronomer (OOM), they probably mean for them to use 1 AU, but it might be good to check.

 

[astronomystudent]

Answers to number 1

Answers:
1.) perihelion = 0.24
aphelion = 23.76
period= 41.57 years (do i need to convert to days?)
this is the equation i found for velocities of aphelion/perihelion:
v(perihelion) = [(1+e)/(1-e)]^1/2
v(aphelion) = [(1-e)/(1+e)]^1/2
thus
v(perihelion) = 9.95
v(aphelion) = .1005

 

[astronomystudent]

Answers To Number 3 &4

ANSWERS:
3.) orbital period (jupiter) = 4332.71 days
convert days ------> years
4332.71 days = 11.8704384 years
p^2 = a^3
(11.8704383 years)^2 = A^3
A = 5.20 A.U. (correct units)??

4.) orbital perid (mars) = 686.98 days
convert days -------> years
686.98 days = 1.882137 years
p^2 = a^3
(1.882137 years)^2 = A^3
A = 1.52 A.U. (correct units)??

I DON'T KNOW ABOUT NUMBER 2. BUT HE SAID IN CLASS, THAT NUMBER 2 WENT ALONG WITH NUMBER 1. SO I GUESS USE THE PERIHELION DISTANCE, B/C THAT WOULD BE THE CLOSEST DISTANCE?

 

[SpaceTiger]

v(perihelion) = [(1+e)/(1-e)]^1/2
v(aphelion) = [(1-e)/(1+e)]^1/2
thus
v(perihelion) = 9.95
v(aphelion) = .1005

Those equations are missing a normalization:

v_p=\frac{2\pi a}{P}\sqrt{\frac{1+e}{1-e}}
v_a=\frac{2\pi a}{P}\sqrt{\frac{1-e}{1+e}}

 

[SpaceTiger]

SO I GUESS USE THE PERIHELION DISTANCE, B/C THAT WOULD BE THE CLOSEST DISTANCE?

Closest distance to what?

 

[astronomystudent]

It Is The "nearest Point To Earth"? I Think? That Is What The Problem Is Looking For... I Can't Solve It Without A Distance.

Are My Other Answers Correct?

 

[SpaceTiger]

It Is The "nearest Point To Earth"?

Decompose the word. What does "helio" refer to?

Are My Other Answers Correct?

Looks right, but you haven't solved 3 and 4 yet, you just found the known values of the planet semi-major axes.

 

[astronomystudent]

Answers:
1.) for the velocities using the new equation:
aphelion = .52009654 A.U. (units)?
perihelion = 18.04673691 A.U. (units)

2.) helio refers to: having the sun as the center?

3,4) what do i do with the resonance? do i multiply by 2 in the first one and then 5 in the second one? I'm confused.

 

[SpaceTiger]

Answers:
1.) for the velocities using the new equation:
aphelion = .52009654 A.U. (units)?
perihelion = 18.04673691 A.U. (units)

What are the dimensions of velocity? Are they distance, time, distance*time, distance/time? Are A.U. units of distance, time, temperature, etc? If you don't know the answer to these things, they can be found on the web or in your textbook.

2.) helio refers to: having the sun as the center?

"Perihelion" refers to the orbit's nearest point to the sun.

3,4) what do i do with the resonance?

What does it mean to be in resonance? For example, what does it mean to be in a 2:1 resonance?

 

[astronomystudent]

1.) velocity = distance/time
thus A.U. is only a measure of distance so I'm stuck again...

2.) it that is the nearest point, than that deals with solar measurements. I am stuck again, so I am still trying to find out what "its nearest point to Earth" is...

3,4) resonance is the circumstance in which two characteristic times are related in some simple way??

 

[SpaceTiger]

1.) velocity = distance/time
thus A.U. is only a measure of distance so I'm stuck again...

Show your calculation then.

2.) it that is the nearest point, than that deals with solar measurements. I am stuck again, so I am still trying to find out what "its nearest point to Earth" is...

Just use 1 AU. As tony said, there isn't enough information to determine it, so you'll have to make an approximation.

3,4) resonance is the circumstance in which two characteristic times are related in some simple way??

The "characteristic times" are the periods. So then how are the periods of two planets related if they're in a 2:1 resonance?

 

[astronomystudent]

1.) I showed all my calculations.

3,4) I don't know?!? I've given you all I got and that definiton was from my book.

2.) I am working on.

 

[SpaceTiger]

1.) I showed all my calculations.

This is all you showed me:

aphelion = .52009654 A.U. (units)?
perihelion = 18.04673691 A.U. (units)

Write out the equations you used and the values of each variable you substituted for and I can help you fix what's confusing you. I'm not going to solve the problem for you.

3,4) I don't know?!? I've given you all I got and that definiton was from my book.

Ok, then we'll have to back up. What is a ratio? If the length of an object is in a 2:1 ratio with the length of another object, what does that mean?

 

[astronomystudent]

Answers 3&4:
this is what i think i do with resonance...
3.) i multiplied the period of Jupiter by 2 and then by 5 for this asteroid.. thus:
period for asteroid = 118.704384 years
A = 24.153 A.U.

4.) i did the same thing just 2:5
thus:
period for asteroid = 18.82137 years
A = 7.076 A.U.

 

[astronomystudent]

2.) he said that the problems 1 &2 go together. I am supposed to use an answer i get from problem one (distances i am guessing) for the distances in 2. does that make sense? is it possible?

 

[astronomystudent]

1.) Most velocity = distance/time
the units should be km/s or m/s
using the equation you gave me for velocity: 2(pi)a/p * square root (1-e)/(1+e) doesn't have those units
the units for a = A.U.
the units for p = years
so that would leave me with A.U/years
distance/time
but that doesn't seem right? should i convert to kilometers and seconds?

 

[SpaceTiger]

Answers 3&4:
this is what i think i do with resonance...
3.) i multiplied the period of Jupiter by 2 and then by 5 for this asteroid..

Good, this is right so far. It should give you:

5P_{ast}=2P_{Jup}

Then you can just solve for the period of the asteroid. The result you got isn't the solution to this equation however. Your result looks like it's just ten times the period of Jupiter.

4.) i did the same thing just 2:5
thus:
period for asteroid = 18.82137 years
A = 7.076 A.U.

Same mistake here.

I am supposed to use an answer i get from problem one...does that make sense? is it possible?

Sounds fine to me.

so that would leave me with A.U/years
distance/time
but that doesn't seem right? should i convert to kilometers and seconds?

AU/years is an acceptable unit, but people are more used to m/s or km/s, so it might be a good idea to convert to that. Good job figuring out the units, by the way.

 

[astronomystudent]

2.) Use the perhelion as my distance than?

3,4) Confused as to what you mean by that? What is my mistake?

 

[astronomystudent]

2.) Thanks!

 

[SpaceTiger]

2.) Use the perhelion as my distance than?

Use 1 AU as you closest distance and the aphelion as your furthest.

3,4) Confused as to what you mean by that? What is my mistake?

You said it yourself. The 5:2 resonance means that five times the asteroid's period is two times Jupiter's period. Do you know how to solve the equation I put up there?

 

[astronomystudent]

Final Answers I Think?

Answers:

1.) perihelion= 0.24 A.U.
aphelion = 23.76 A.U.
period = 41.57 years
velocity perihelion = 85.61 km/s
velocity aphelion = .865 km/s

2.) I think I use the perihelion for the distance but I am not sure? I haven't started any work on this problem.

3.) orbital period Jupiter = 4332.71
convert to years = 11.8704383
period asteroid = 4.75 years
semi-major axis asteroid = 2.82 A.U.

4.) orbital period Mars = 686.98 days
convert to years = 1.882137
period asteroid = 4.71 years
semi-major axis asteroid = 2.81 A.U.

 

[astronomystudent]

2.) I used 1 A.U. as the closest distance, and the Aphelion= 23.76 A.U. as the furthest distance.

Using the inverse square law, I found the answer to be: 0.0018 (How bright the comet would appear at its maximum distance)? Is that correct? Units for luminousity or is it just brightness.

 

[SpaceTiger]

Everything looks good to me.

Is that correct? Units for luminousity or is it just brightness.

The teacher didn't give you any units in the problem, so just the number is fine.

 

[tony873004]

Would it make more sense to use 1 AU - 0.24 AU for 0.76 AU, since the 1 AU is Earth / Sun distance and the 0.24 is comet / Sun distance. So if Earth were aligned for closest approach to the comet at the comet's solar perihelion it would be 0.76 AU away.

 

[SpaceTiger]

Would it make more sense to use 1 AU - 0.24 AU for 0.76 AU, since the 1 AU is Earth / Sun distance and the 0.24 is comet / Sun distance. So if Earth were aligned for closest approach to the comet at the comet's solar perihelion it would be 0.76 AU away.

Why would it be aligned for closest approach? That seems as arbitrary as any other choice.

 

[tony873004]

Why would it be aligned for closest approach? That seems as arbitrary as any other choice.

Just because at perihelon, the comet is 0.76 AU from Earth's orbit, assuming they're in the same plane. So using 0.76 AU seems a little less arbirtary to me than to use the Earth / Sun distance as the Earth / Comet distance. But you're right, any guess is just an arbitrary guess.

 

[Chronos]

Hmm, the simple answer looks pretty good. It was a pretty simple question to begin with. ST's "Good, this is right so far." is correct. It should give you:

 

[RedScout]

Is this math typical to astronomy or is it more complicated? I'm wondering because I'm considering astronomy as a future career.

 

[SpaceTiger]

Is this math typical to astronomy or is it more complicated? I'm wondering because I'm considering astronomy as a future career.

I think this is typical of an intro level astronomy course in college (or high school). Graduate students would be expected to quickly do this sort of calculation in their head (or at least obtain an order of magnitude).

For an observational research paper, you'll be expected to do calculations roughly at the level of the ones I do here:

https://www.physicsforums.com/showthread.php?t=102126"

Even more mathematical sophistication would be expected of theorists.

 

[RedScout]

Thanks for the help and information. Would it be safe to say it is advanced algebra on an astronomy scale? I'm in 10th grade (in America), what classes would you reccomend to prepare for college? The school that I go to dosen't offer an introductory to Astronomy class. Although, it did teach about Astronomy in a physical science class I had last year but we didn't reach it.

 

[SpaceTiger]

Thanks for the help and information. Would it be safe to say it is advanced algebra on an astronomy scale?

No, you definitely need calculus. Look at some of my later posts in the thread. From what I remember, the major I took in astronomy required the following undergrad math classes:

- Calc I and II
- Vector Calc
- Linear Algebra
- Differential Equations

If you want to do any general relativity, you'll need to learn tensor manipulation and basic differential geometry (sometimes done in the GR course itself). Hardcore theorists would also benefit from complex analysis and group theory.

By the time you graduate high school, you should, at the least, have taken algebra II and classical mechanics (intro physics). Preferably, you would have some exposure to calculus and electromagnetism as well.

 

[RedScout]

I'll need to work hard it seems.

I've been reading general astronomy books to try and get a good understanding of everything as a whole before I start learning more specific things. Hopefully it will come to good use in the future.

Thanks for your help again.