# Hands-On GW Astronomy - Student Worksheet Dan G.
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### Here are some other problems

Message 68127 in response to message 68124

Here are some other problems I found with the outreach exercise paper:

Problem #8 calculates k the evolution constant. I've confirmed their value of 5e-6, but when they use it in the formula for orbital period for the binary system, and solve for t, something goes wrong. They get 4.8e11 or about 15,000 years. My value isn't even close. Can you confirm?

Problem #9 calculates a new value of k for the Last .1 second before coalescence. Their value is k=5.2e-6 but I get k=8.2e-7. This is important because they use this k value in Problem #10 to for a calculated chirp mass (instead of an estimated value based on Chandrasekar limit of 1.4 solar mass for a NS). Can you confirm?

_dan Chipper Q
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### RE: Problem #8 calculates k

Quote:
Problem #8 calculates k the evolution constant. I've confirmed their value of 5e-6, but when they use it in the formula for orbital period for the binary system, and solve for t, something goes wrong. They get 4.8e11 or about 15,000 years. My value isn't even close. Can you confirm?

Well, I came up with a tad over 24 trillion years.

Here are some of the numbers I was using:

Mass of sun = 1.98892 e30 kg
1.4 x Mass of sun = 2.784488 e30 kg
G = 6.67 e-11 cubic meters per kilogram per second per second
c = 3 e8 meters per second
Chirp Mass = (((2.78e30)^2)^(3/5))*(((2*(2.78e30))^(-1/5)) = 2.42 e30 kg

96/5c^5= 7.9 e-42
(2Ï€)^8/3 = 134.4
(GM)^5/3 = [( 6.67 e-11)( 2.42 e30 )]^5/3 = 4.79 e33
then
k â‰¡ (7.9 e-42)(134.4)(4.79 e33) = 5.1 e-6

P_0 = 1000 seconds, and 1% of that is 10 seconds, so the problem is to find t when P_orb(t) is 1010 seconds.

To solve for t, I used:
t = 3 ( P_0^8/3 â€“ P_orb(t)^8/3) / 8 k,

and came up with

t = 7.57352941176471e+20 seconds, or
t = 24 082 946 738 662 years.

I'll try to work on #9 & #10 tomorrow...

edit: One thing I noticed, when using a scientific calculator (like this one that I used), you have to be very careful how you enter the numbers. In the calculation for the Chirp Mass (above), I kept getting wrong answers until I finally used as many parentheses as you see there. Obviously the order in which the operations are performed is critical for arriving at the correct answer, and the calculator needed some help in parsing that order. Chipper Q
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### Well, before I get to 9 & 10,

Well, before I get to 9 & 10, let me redo #8 with P_orb(t) = 990 seconds, duh!

- - - edit - - -

Okay, redoing #8 using P_orb(t) = 990 seconds, since the orbital period decreases by 1% from P_0 = 1000 seconds, I come up with:

t = 2.2 e11 seconds, or
t â‰ˆ 6971.2 years Chipper Q
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### RE: Problem #9 calculates a

Quote:
Problem #9 calculates a new value of k for the Last .1 second before coalescence. Their value is k=5.2e-6 but I get k=8.2e-7. This is important because they use this k value in Problem #10 to for a calculated chirp mass (instead of an estimated value based on Chandrasekar limit of 1.4 solar mass for a NS). Can you confirm?

For #9, using k = 3(P_0^8/3 â€“ P_orb(t)^8/3) / 8 t,
I come up with k â‰ˆ 8.2 e-7

Then, for #10, solving for M and coming up with: M = ( [5kc^5 / 96(2Ï€)^8/3 ] ^ 3/5 ) / G,
I get for the calculated Chirp Mass (using the value I got for #9 above, k = 8.2 e-7):

M = 8.1 e29,

which looks fairly close... (within a factor of 10, anyway)

How's that?

- - -

edit:

When I made the mistake with #8, I should have noticed that P_orb(t) > P_0, and my answer of 24 trillion years should have had a minus sign in front of it, which I think is pretty cool, since a negative value for the time just means that it happened before the time at P_0. I wonder if this is helpful or used to constrain models of how binary systems form, since you can determine the orbital period when the system formed if you know the age of the stars. Am I interpreting this correctly? Dan G.
Joined: 10 Feb 07
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### HA! Thank you. That's where I

Message 68131 in response to message 68129

HA! Thank you. That's where I went wrong eariler. Of course, when considering an inspiral binary system, the period decreases, so 1% of the orbital period P_orb(t) original value of 1000s would be 990s NOT 1010s!

But I think you have a problem with your solve-for-t equation. I think it should be:

t=-3(P_orb(t)^8/3 - P_0^8/3)/8 k

Then you get t=2.0e11 sec or 6,367 years. Notice that the sign of t is positive now.

Still, this is different than the solution given, t=4.8e11 or 15,000 years. Chipper Q
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### RE: But I think you have a

Message 68132 in response to message 68131

Quote:

But I think you have a problem with your solve-for-t equation. I think it should be:

t=-3(P_orb(t)^8/3 - P_0^8/3)/8 k

Then you get t=2.0e11 sec or 6,367 years. Notice that the sign of t is positive now.

Still, this is different than the solution given, t=4.8e11 or 15,000 years.

They're actually equivalent statements. I factored a -1 out of the numerator, sort of. Note that you can multiply the -3 by -1 if you also multiply the quantity inside the parentheses by -1. Doing this twice is the same as multiplying by (-1)(-1) = 1. Check that in my version the order of P_0 and P_orb(t) is reversed from yours. You'll get a negative value for t when P_orb(t) > P_0 either way. This makes sense when you consider, as we both learned well, that the orbital period decreases as time progresses, and P_O is the zero point on the time line.

But you're right about our answers being closer to each other than either is to the answer provided. That was a good problem! :) Dan G.
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### For #9 I'm getting k=7.97e-7,

Message 68133 in response to message 68130

For #9 I'm getting k=7.97e-7, but their value is 5.2e-6 which is signficantly different from ours.

Then for #10, I get the same value for chirp mass as you do, M=8.1e29. Their value of M=2.5e30 is very close to the estimated value they calculated earlier in Problem #5 M=2.4e30. So I believe they were trying to demonstrate how estimated vs. calculated were very close. But as you and I have seen, this is not the case.

I'm concerned why we've seen so many issues with the calculations. The purpose of this paper is excellent in using it as an outreach vehicle, but with a lot of errors, it doesn't serve to shed the proper light on the science. Just my opinion. I guess it should have been uploaded to a pre-print server first, and then officially published.

_dan

Quote:
Quote:
Problem #9 calculates a new value of k for the Last .1 second before coalescence. Their value is k=5.2e-6 but I get k=8.2e-7. This is important because they use this k value in Problem #10 to for a calculated chirp mass (instead of an estimated value based on Chandrasekar limit of 1.4 solar mass for a NS). Can you confirm?

For #9, using k = 3(P_0^8/3 â€“ P_orb(t)^8/3) / 8 t,
I come up with k â‰ˆ 8.2 e-7

Then, for #10, solving for M and coming up with: M = ( [5kc^5 / 96(2Ï€)^8/3 ] ^ 3/5 ) / G,
I get for the calculated Chirp Mass (using the value I got for #9 above, k = 8.2 e-7):

M = 8.1 e29,

which looks fairly close... (within a factor of 10, anyway)

How's that?

- - -

edit:

When I made the mistake with #8, I should have noticed that P_orb(t) > P_0, and my answer of 24 trillion years should have had a minus sign in front of it, which I think is pretty cool, since a negative value for the time just means that it happened before the time at P_0. I wonder if this is helpful or used to constrain models of how binary systems form, since you can determine the orbital period when the system formed if you know the age of the stars. Am I interpreting this correctly? Chipper Q
Joined: 20 Feb 05
Posts: 1,540
Credit: 708,571
RAC: 0

### RE: For #9 I'm getting

Message 68134 in response to message 68133

Quote:

For #9 I'm getting k=7.97e-7, but their value is 5.2e-6 which is signficantly different from ours.

Then for #10, I get the same value for chirp mass as you do, M=8.1e29. Their value of M=2.5e30 is very close to the estimated value they calculated earlier in Problem #5 M=2.4e30. So I believe they were trying to demonstrate how estimated vs. calculated were very close. But as you and I have seen, this is not the case.

I'm not sure how to judge how far off our answer is from the one given. If you multiply ours (7.97 e-7) by a factor of just 6.53 then it's about right on, so we're well within one order of magnitude, which I think is not too shabby! :) It's also possible we were sloppy with the proper number of significant digits, e.g., using 3,000,000 m/s for the speed of light instead of 299,792,458 m/s. Note that a slight difference here gets raised to the fifth power in the calculation...

And in the case for the calculated Chirp Mass (#10), you only have to multiply our answer (8.1 e29) by a mere factor of 3 to arrive at the given answer. We were way less than one decimal place off, out of thirty! I think we were â€œastronomically closeâ€? to the proper answer, considering the astronomical size of the numbers. :)