Find Absolute Magnitude, distance and "T" of binary star system

[Unteroffizier]

This is the problem given at my basic astronomy course. If I turn it in within a month correctly, I get 10 extra points in the finals.

Problem:
We have binary star system consisting of Star A and Star B. Astronomers have observed it for 11 years, during which it has moved from point A to point B. The major axis "a" = 4.5 degrees.

Relative magnitudes of A and B are 3.9 and 5.3 respectively. The system is being observed from a right angle (from the top, which means no transits, no magnitude shifts and no Doppler effect).

The orbits were converted into a simpler, Keppler-like system beforehand, where the motion of Star B is relative to Star A. Therefore, only Star B is in (relative) motion to simplify.

Find:
a) The orbital period of B star.
b) Absolute magnitude.
c) Distance from observatory.Relevant Equations
Kepler's Second Law
Looping mass equation
Trigonometric functions (tg)
Area equation (ab*(arccos*h/a - h/a^2*sqrrt(a^2 - h^2)))
M = m + 5 - 5log (d)

Solution Attempt
I have not yet done the mathematics required to come up with valid results, but here is how I wish to continue. I only wish to know if I have the right idea, and if not, what I should change, so that I do not waste time, as I must focus on my mechanical engineering high school course as well.

a) Find absolute magnitude using "M=m+5-5log(d)" where m=ma+mb. I do not know if relative magnitudes of both stars add together to form one relative magnitude, or if ma = m. Need clarification

b) Knowing absolute magnitude, find distance. If absolute magnitude = "n" at "d=10pc", then it should be simple mathematics to find "d."

c) Use tg of 4.5 and "d" to find physical length of apastron "a", the major axis.

d) Use distance between two points Star B traveled to calculate area in relation to 11 years and use Kep. 2. law to find "T" period.

e) Write excel program to continually calculate mass of stars, where Input 1 assumes that total system mass M = 2 solar masses, slowly gaining more precise results with every cycle. I do not have the equation cycle on hand, but I will find it eventually.

We are technically not meant to find the masses of the stars, but he gave us said equation, and told us to use it, and so I will.

Am I on the right track? Any fallacies?

 

[collinsmark]

'Sounds like an interesting problem. But this part lost me:

We have binary star system consisting of Star A and Star B. Astronomers have observed it for 11 years, during which it has moved from point A to point B. The major axis "a" = 4.5 degrees.

Is there a figure missing or something? I don't know what point A and point B are, or how they fit into the problem.

You mention the major axis is given in units of degrees. The major axis of an ellipse is a measure of length (distance). I suppose this "distance" could be expressed as an angular distance as viewed from the observatory. But holy-moly, 4.5 degrees is about 9 full moon [angular] diameters as viewed from Earth! That doesn't make sense to me if we are observing from this Earth. So I think I'm missing something or misunderstanding something.

Can you elaborate on that part?

 

[phyzguy]

I find it difficult to understand the problem. Is there a diagram you did not include?

 

[collinsmark]

Also,

e) Write excel program to continually calculate mass of stars, where Input 1 assumes that total system mass M = 2 solar masses, slowly gaining more precise results with every cycle. I do not have the equation cycle on hand, but I will find it eventually.

I'm not following that at all.

By the way, what's this about M = 2 solar masses? Where does this information come from? Are we missing some information here?

We are technically not meant to find the masses of the stars, but he gave us said equation, and told us to use it, and so I will.

What equation is this?

I think it likely that in order to solve this problem you may need some method of estimating the star system's mass that does not depend on the distance from the observatory. Is there any additional information about the stars, such as their spectral properties or some-such, that would allow someone to estimate what types of stars they are? Does this relate to the equation you mentioned above?

 

[Unteroffizier]

Thanks for your response! Ah yes, that's the confusing part alright. Here's how it works:

It's a set of equations, in which the input assumes that both stars are approximately the mass of Sol, because 90% of stars in the universe are within the range of 0.1 to 10 solar masses. Therefore, we assume that the binary system has a mass of 2 Suns. Now, here's the interesting part.

The output of this equation is, in fact, a more accurate estimate of the total system's mass! Now, here's what you do next. You take that more accurate estimate and replace 2 solar masses with it. You get an even more accurate estimate. You keep repeating this 'till the differences between input/output are smaller than 1%, in which case you have a highly accurate estimate of the system's mass.

It is called the dynamic parallax method, apparently.

Anyway, since it's being viewed from a 90 degree angle, the system cannot be measured by means of Doppler effects, spectral shifts, transits, or any of that. It's meant to be done purely mathematically.

So, do you think the method above should get me good results? I'm doing it more for fun than anything, if I am to be honest!

 

[Unteroffizier]

Oh damn, I did not notice your original comment. Yes, there is a diagram missing, and yes, it is meant to be solved using trig. I think it may have been meant to be 4.5 angular seconds?

I will send you a link to the diagram and the whole problem here. It is in Czech though. Diagram should be understandable however. If it wants a password, it is "20kurs16"

The password is a formality, the lecturer even told us that it is fine to share with others.

http://observatory.cz/studijnimaterialy1?pswd=20kurs16

Go down to "Studijní materiály" and click on the "seminarni prace" PDF, second from bottom.

@collinsmark
@phyzguy

 

[collinsmark]

Ah, Dynamical Parallax. A quick Internet search comes up with some useful explanations and examples. Dynamical Parallax has been used by astronomers for quite awhile. Neat. I learned something today.

That sounds like a fun project.

Btw, I'm guessing that angular distances measured is 4.5 arcsec rather than degrees.

1 degree = (1 degree) (60 arcmin/degree) (60 arcsec/arcmin) = 3600 arcsec

So yes, it seems to me that you can use the Dynamical Parallax method to estimate answers to this problem.

I'm not sure how much help you can get from Physics Forums (PF) though, without more detailed attempts at your solution, including all the relevant formula and algorithm given to you by your instructor. Although you might be able to break things into smaller bits.

But I must say, I hope things go well!

 

[collinsmark]

I'm doing my best to translate the "seminarni_prace_1516.pdf" link that you gave earlier. I don't think that the semi-major axis of the ellipse is 4.5 arcsec.

Rather the 4.5 arcsec is just the "half-moon"-like shape swept out as the orbiting star goes from one side of the focus point to the other side, at this particular segment of their orbit during the 11 years. The shorter length of this half-moon-like shape is 3.4 arcsec. The area swept out during this eleven year period is indicated by the dark blue area in the next figure.

[Edit: Oh wait. Nevermind. I think I was just mistranslating. Now that I look at it again I think the semi-major axis of the ellipse is a = 4.5 arcsec and the semi-minor axis is b = 3.4 arcsec. Sorry about that. You'll still need to do some work to find the orbital period though.]

Your instructor gives you an equation that you can use to calculate the area area swept out given the dimensions of the dark blue, half-moon-like shape. Only after that can you calculate the area of the whole ellipse and thus the orbital period.

So the first stage of your project is to calculate the orbital period from that information.

 

[Unteroffizier]

Ay, nice man. Thanks so much. I am only in my first year of high school so I have not yet gotten into arc trig. I know sin/cos/tg etc., but Arccos is still a mystery. Will check that out.

Anyway, you seem to be right. I read it all wrong. I do have a formula for the area. Unfortunately, it contains arccos, so I will have to ask around my school for some assistance. After that, I can use Kep. 2nd law to find the orbital period. Absolute magnitude is easy and I can even find the distance with that, and I will use dynamical parallax for the mass.

Alright, that about does it! Thanks much for your help, I appreciate it. Hopefully I can get this assignment done and finally become a member of the Prague Astronomy Group! That would be quite amazing!

 

[collinsmark]

Ay, nice man. Thanks so much. I am only in my first year of high school so I have not yet gotten into arc trig. I know sin/cos/tg etc., but Arccos is still a mystery. Will check that out.

Anyway, you seem to be right. I read it all wrong. I do have a formula for the area. Unfortunately, it contains arccos, so I will have to ask around my school for some assistance. After that, I can use Kep. 2nd law to find the orbital period. Distance should be possible afterwards by means of major axis and basic trig, absolute magnitude is easy and I will use dynamical parallax for the mass.

I just edited my previous post. On second thought, I think you may have been right in that the semi-major axis of the ellipse is a = 4.5 arcsec and the semi-minor axis is b = 3.4 arcsec.

It was a translation problem on my part. Sorry about that.

But you'll still have to figure out what fraction of the area that 11-year segment corresponds to, so there's still some work for you to do in finding the orbital period.

Regarding arccos(): That's is simply the inverse cosine.

$\cos \theta = \frac{\mathrm{adjacent}}{\mathrm{hypotense}}$

$\mathrm{arccos} \left( \frac{\mathrm{adjacent}}{\mathrm{hypotense}} \right) = \theta$

Be sure to work in radians (rather than degrees) for this project.

Alright, that about does it! Thanks much for your help, I appreciate it. Hopefully I can get this assignment done and finally become a member of the Prague Astronomy Group! That would be quite amazing!

 

[Unteroffizier]

Alright, good. I will get to work tomorrow. I still have over a month left, but I want to be one of the first people to turn it in.

Again, thanks a lot. I was a bit lost in all that text.