There is a Cepheid variable located in a nebula very far away (parallax measurement is not an option). It illuminates the nebula and, in particular, a knot of gas within the nebula some 16.01 arcseconds away from it. The pulsing of the knot of gas is 185 days out of phase from the star. How far away is the star?
You get credit if you get the answer everyone else in my class got and bonus points if you get the correct answer.
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I don't know if it's possible to figure this one out, at least not with the basic math that I still remember. I'm guessing that your entire class assumed the glowing knot of gas was 185 light days to the side of the star at a roughly right angle and just worked the trig from there. If it isn't, and in reality it probably isn't since the star could be angled closer or further away from the glowing gas instead of side by side at the exact same distance from Earth, it would screw the angle & phase relationships beyond the point of my limited math abilities to solve it.
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I don't know if it's possible to figure this one out, at least not with the basic math that I still remember. I'm guessing that your entire class assumed the glowing knot of gas was 185 light days to the side of the star at a roughly right angle and just worked the trig from there. If it isn't, and in reality it probably isn't since the star could be angled closer or further away from the glowing gas instead of side by side at the exact same distance from Earth, it would screw the angle & phase relationships beyond the point of my limited math abilities to solve it.
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I'm pretty sure that even if it isn't at a right angle, you treat it as if it were, because the component of the distance between the star and the knot that is parallel to the axis between the star and earth would not affect the relative timing at all since light travels at a constant speed. In other words, if the nebula were almost on a straight line between the star and the earth, then no matter how far apart they are they would appear to light up at the same time, since light travels at c.
Unless the knot of gas is behind the star... Actually, the Cepheid variable and the knot could be 185 light-days away from each other, or 185 light-days plus some integer*c*frequency.
If I understand it right, 1" is the arc subtended by a 1AU object at 1parsec (206265 AU). So if we've got 9.01" and objects that are 181 light-days (~31300 AU) apart, and if I've done my math right... that's 11300 LY. My guess is that's the wrong answer, and that I'm not accounting for the close coincidence between the period of the "flash" and the time it takes the Earth to traverse half its orbit. But I'm not getting how to account for that.
Through some abuse of trigonometry and the assumption that the line between the star and the cloud of gas is perpendicular to the bisecting line to earth, forming an isosceles triangle, then the distance should be on the order of 92.5 light-days / sin(0.002224) = 2.383e6 light-days, or 6524.3 light years. This is only true if the assumption is valid. If the triangle is some non-regular shape, then the phase of the period is necessary in order to add another term to the equation in order to normalize it into a isosceles triangle and solve for the corrected distance.
You're all on the right track, but nobody has gotten the extra credit. Except Morilore, who didn't actually provide a worked-out solution.
aerius wrote:Spoiler
I don't know if it's possible to figure this one out, at least not with the basic math that I still remember. I'm guessing that your entire class assumed the glowing knot of gas was 185 light days to the side of the star at a roughly right angle and just worked the trig from there. If it isn't, and in reality it probably isn't since the star could be angled closer or further away from the glowing gas instead of side by side at the exact same distance from Earth, it would screw the angle & phase relationships beyond the point of my limited math abilities to solve it.
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You can use a small-angle approximation in this case because the star is too far away for parallax measurements of distance. In other words, regardless of the actual distance between the Cepheid variable and the knot of gas, it's going to be so incredibly small compared to the distance of both from the Earth that we can assume the distance from the Earth is the same to both - and that gives us a picture with an isosceles triangle, the Earth at the tip, and the star and knot of gas at the base of each leg. The key is to make a second approximation based on small angle about the length of the base of the triangle. That should get you the solution (without any trig!), with one caveat.
Morilore wrote:Spoiler
I'm pretty sure that even if it isn't at a right angle, you treat it as if it were, because the component of the distance between the star and the knot that is parallel to the axis between the star and earth would not affect the relative timing at all since light travels at a constant speed. In other words, if the nebula were almost on a straight line between the star and the earth, then no matter how far apart they are they would appear to light up at the same time, since light travels at c.
Unless the knot of gas is behind the star... Actually, the Cepheid variable and the knot could be 185 light-days away from each other, or 185 light-days plus some integer*c*frequency.
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On your first point: close, but not quite. Like I said to aerius above, you treat it as a right angle because the whole nebula is so goddamned far away and you're trying to find the distance to the star. It doesn't matter if you're off by a term on the order of the radius of the nebula because that's a tiny error.
Second point: very good! Now just put that with a solution and guess what assumptions the class made about your observation.
Turin wrote:Spoiler
If I understand it right, 1" is the arc subtended by a 1AU object at 1parsec (206265 AU). So if we've got 9.01" and objects that are 181 light-days (~31300 AU) apart, and if I've done my math right... that's 11300 LY. My guess is that's the wrong answer, and that I'm not accounting for the close coincidence between the period of the "flash" and the time it takes the Earth to traverse half its orbit. But I'm not getting how to account for that.
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Your assumption about angles is wrong. The angle I'm referring to is simply the measurement of angle on the celestial sphere; it doesn't matter how wide it is or what the radius of the sphere is. But you're on the right track. You can assume that the period of the Earth's orbit is negligibly small compared to the period of the Cepheid variable, or (another way of looking at it) because parallax measurements of distance don't work, the star is so far away from the Earth we may treat the Earth as stationary.
Razaekel wrote:Spoiler
Through some abuse of trigonometry and the assumption that the line between the star and the cloud of gas is perpendicular to the bisecting line to earth, forming an isosceles triangle, then the distance should be on the order of 92.5 light-days / sin(0.002224) = 2.383e6 light-days, or 6524.3 light years. This is only true if the assumption is valid. If the triangle is some non-regular shape, then the phase of the period is necessary in order to add another term to the equation in order to normalize it into a isosceles triangle and solve for the corrected distance.
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You're arriving at pretty much the correct answer with pretty much the correct method. It's possible to justify your assumption about the triangle (hint: 0.002224 ~ 0). You're missing the caveat, however.
And all of science is pretty much educated guesswork
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I assumed that the gas and star were on a line perpendicular to our line of sight, then made an isoceles triangle. Do the trig, and you get (185 ld)/(2*sin(9")) ~ 6500 ly. It doesn't much matter if they're not, except for one case I'll get to in a second, because 9" is pretty much 0, and the small angle approximation holds. If the gas is directly behind the star, the angle actually is 0, and this breaks down, making the problem impossible with the given information. However, if you just give me the period of the variation, I can give you distance to the star no matter where the knot is, because the star's a Cepheid.
By caveat, do you mean that little detail where the cloud could be anywhere on a circle with the star at the center, and thus the actual phase difference between the star and the cloud could be anywhere from 185 days(perpendicular to line to earth) to ~92.5 days(almost directly behind the star, so 92.5 days from the star to the cloud, then an additional 92.5 days for the wavefront from the cloud to get back to the star's distance, by which point it is now 185 days behind the star's wavefront)? In the first case, it'd be as assumed above, about 6500 ly. In the second case, It could be next door, depending on how close to 'behind the star' you want to get. By my guess, the relevant equation here would be similar to d (ly) = (92.5 - 46.25*cos(x)) / 0.014245), where x = 0 degrees for directly behind the star and x = 90 degrees for perpendicular. This sets a minimum distance of ~3250 ly, which you'd never see since the cloud would be directly behind the star at this point. Expanding this to be in front of the star would take more thinking than I can sustain at 2 am, so I'll shelve that for later. I'm probably a bit inaccurate in assuming how the location on the circle affects the phase of the cloud in relation to the star, tho, since I'm just using a straight up cosine.
Your assumption about angles is wrong. The angle I'm referring to is simply the measurement of angle on the celestial sphere; it doesn't matter how wide it is or what the radius of the sphere is. But you're on the right track. You can assume that the period of the Earth's orbit is negligibly small compared to the period of the Cepheid variable, or (another way of looking at it) because parallax measurements of distance don't work, the star is so far away from the Earth we may treat the Earth as stationary.
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I'm obviously mucking up my geometry somewhere, but I thought that if I project two lines to an infinite sphere that subtend 1", those lines intersect the end points of the 1AU line at 1 parsec distance from the origin? That makes the problem just a simple similar triangles problem.
But obviously not, because I'm not getting the same answer if I just use trig. I can assume the two objects are at the same distance from Earth, because 181LD in any direction is going to be swamped by whatever distance we come up with. So that's just sin(9.01") = 92.5 / x, or about 6500 LY.
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The thing you're forgetting here is that the distance between the two may NOT be 185ld, but it IS 16.01". So, if we say that the base of a triangle perpendicular to us with the star at one vertex, and the cloud at the end of the hypotenuse which angles towards or away from us, then you have to move the location of the star as you change the angle of the hypotenuse in order to keep the base at 16.01". I've made an image of how I view this problem HERE.
Why are we trying to measure the distance using the pulsing of the nebula anyway, rather than the length of the star's pulsation cycle? Aren't Cepheid variables the stars in which the variability period of is proportional in some way to the mass/actual luminosity? So shouldn't we just be able to measure the luminosity as seen from earth, measure the length of the variability cycle and then do our funky maths or look up the thing on our Cepheid variable chart and then be able to figure out from that what the star's distance is?
Anyway, for what I did, the star is on a circle with a circumference of 185*3600*360/16.01 light years. I took the radius of that: sqrt (185*3600*360/16.01/pi) and got 2180light years (to three significant figures). I figured that with the distance between the star and the nebula being only 185ly, any differences in the distance between the star and Earth and the nebula and Earth would be negligible.
I fixed your spoilers. The first tag needs to have an =, even if you don't include a label, otherwise it doesn't work. So Spoiler
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Assumption: The star and the nebula are at the same distance r from earth, separation is d.
Trigonometry gives us d/2=r*sin(alpha/2), and solving for r we get
r=d/(2*sin(alpha/2)). d equals 185/365.25 LY, plugging in yields
r=6525.5 LY
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Previously I assumed that the line between star and knot is perpendicular to the line of sight. Can we estimate what happens if we wiggle that line by an angle phi?
Some BOTE approximation seems to tell me that if we move the knot on its LOS till the angle earth-star-knot is 90°+phi, the light has to make a detour not of d, but of d*(tan(phi)+sqrt(1+tan^2(phi))). I'll assume that phi is within +- 0.1 rad, and take the average of the extremes. The result suggests that we overestimated the distance by 0.5%, giving now r=6493 LY.
However, it also suggests that our value cannot be more accurate than about 1%, so we better state that as
r=6493 +- 65 LY, blatantly ommiting any other contributions to the error bar.
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The description "Cepheid within a nebula" fits AFAIK only on RS Puppis and here its distance is given as 6500 LY, with an error of 90 LY.
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Yes, it's a Cepheid with a close correlation between luminosity and period. But in order to exactly determine that correlation, we must measure the distance to one Cepheid using another method, to calibrate the Cepheid method.
Last edited by Glass Pearl Player on 2009-11-05 10:44am, edited 1 time in total.
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It seems like this should be pretty straight-forward, though I suspect there'll be a twist.
It takes 185 days for the light from the star to reach the knot of gas in the nebula, meaning the star is 185 light-days from the knot of gas. The knot of gas is 16.01 arcseconds from the star, or 16.01/3600 degrees.
I assume the following:
We are to ignore gravitational lensing artifacts
we may presume that the star and the knot of gas are co-radial with respect to Earth.
The 185 light-day distance will cut through the radius of the arc circle, so we can't just measure along the circumference and call it good. Instead, we'd have to let the triangle formed by the distance between Earth and the Star (A), the distance between Earth and the Knot (B), and the distance between the Knot and the Star (C) describe the relationships. The angle between A and B (α) is our arc measurement, or 16.01/3600 degrees.
To properly establish the distance, we need to bisect the ABC triangle through α to create a pair of right triangles which we can operate on with simple trig. α becomes α' and α'', both of which measure 16.01/7200 degrees. The "base" (perpendicular to Earth) of each of these triangles is 185/2 light days. This is sufficient information to solve for the distance between the star and knot of gas (again, assume they're co-radial, with respect to Earth).
sin(α') = (185/2 light days)/r
r = (92.5 light days)/sin(α') ≈ 2,383,447 light days ≈ 6,525 light years
16.01" is ~7.75e-5 radians. With an arc length of 185LD, that's a radius of ~6500LY. Of course, I re-read my original answer and for some reason I put 9.01" and 181LD in, which is obviously not going to generate the right number no matter what I do.
Not sure how to close this, or if it can be closed but here's my guess.
D(lambda, T) = 6.1x10^21 cm + 3.8x10^14 cm/s*(lambda*T)
where lambda is how many pulses out of phase the knot and the star are and T is the period of the cephiad variable.
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Oops, looks like I broke the spoilers. Could a mod please fix?
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"Liberals tend to clump together in places where they can avoid reality and diversity of opinion, like big cities, especially in the east and west coast and college towns." --nettadave2006
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Fixed it. See my edit of Lus's post for an explanation of how spoiler tags work.
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Surlethe wrote:Fixed it. See my edit of Lus's post for an explanation of how spoiler tags work.
Sorry, didn't read the rest of the thread until after I posted. Thanks for the fix though.
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I think Surl's pulling a fast one on us. A period of 185 days gives us an absolute magnitude of -6.37 or less (more negative) for the Cepheid, which is pushing the absolute limit of how bright these things get. As such, why do we assume that the Cepheid's period is at least 185 days? Why can't it be cycling faster and the knot is reflecting light from the Cepheid that was emitted 18 cycles ago (and thus, 185 days = 5 days + 18 periods * 10 days/period — the real phase difference is 5 days out of phase)?
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I think Surl's pulling a fast one on us. A period of 185 days gives us an absolute magnitude of -6.37 or less (more negative) for the Cepheid, which is pushing the absolute limit of how bright these things get. As such, why do we assume that the Cepheid's period is at least 185 days? Why can't it be cycling faster and the knot is reflecting light from the Cepheid that was emitted 18 cycles ago (and thus, 185 days = 5 days + 18 periods * 10 days/period — the real phase difference is 5 days out of phase)?
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Why would either assumption change the distance to the star, if we use only the information given in the original problem statement? I read that as saying the gas is about 185 light-days away from the star, and then used trig and the small-angle approximation.
I think Surl's pulling a fast one on us. A period of 185 days gives us an absolute magnitude of -6.37 or less (more negative) for the Cepheid, which is pushing the absolute limit of how bright these things get. As such, why do we assume that the Cepheid's period is at least 185 days? Why can't it be cycling faster and the knot is reflecting light from the Cepheid that was emitted 18 cycles ago (and thus, 185 days = 5 days + 18 periods * 10 days/period — the real phase difference is 5 days out of phase)?
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Surlethe says that the cloud is 185 days out of phase with the star, nothing about the actual period of the star itself. Your assumption is thus flawed, as well as the idea about multiple cycles. If you go with the multiple cycles idea, it could be cycling any number of times in that 185 days, which makes it useless as a estimation of distance between the cloud and star, at which point you have absolutely no information with which to solve the problem.
Why would either assumption change the distance to the star, if we use only the information given in the original problem statement? I read that as saying the gas is about 185 light-days away from the star, and then used trig and the small-angle approximation.
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Surl doesn't say that. He says that the knot's pulses are 185 days out of phase with the star, not that the knot is 185 light-days from the star. This is an important distinction.
Razaekel wrote:Spoiler
Surlethe says that the cloud is 185 days out of phase with the star, nothing about the actual period of the star itself. Your assumption is thus flawed, as well as the idea about multiple cycles. If you go with the multiple cycles idea, it could be cycling any number of times in that 185 days, which makes it useless as a estimation of distance between the cloud and star, at which point you have absolutely no information with which to solve the problem.
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Why do you assume that Surl has given us enough information to solve the problem? Again, Surl doesn't say that the knot is 185 light days from the star, just that it is 185 days out of phase with the star. But that doesn't tell us what we need to know. 180° out of phase is also 540° out of phase and also 700° out of phase.
In Surl's response to Meliore, he said that she was on the right track in considering how many cycles actually pass before the knot reflects the star's light, in that (star to knot transit time) = 185 days + n * (period of star) for unknown n. I realized, for all we're told, the equation could look like this:
185 days = (star to knot transit time) + n * (period of star)
By the way, I messed up my magnitude calculation: 185 days corresponds to an absolute magnitude of -7.80, which is way beyond the limit of around -6 magnitudes. I'm even more convinced that Surl's pulling a fast one, and the problem is unidentified unless we have the period of the Cepheid — which we ought to have since we're looking at the damn thing.
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You're incorrect in your magnitude calculations - the 185 days is the phase difference, not the period.
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The star in question cannot be any other than RS Puppis, and here is the paper about measuring its distance. It also states the Period as P=41.4 days. This is much shorter than the stated phase shift of 185 days, and the paper does discuss how to resolve the ambiguity.
Also, some more
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Assume for the period a value of P=41.4 days. Take the small angle approximation
r=dT*c/alpha
and dT=dt+N*P, with N unknown.
Therefore, the candidate solutions are spaced P*c/alpha apart, or 1460 LY. If we have another measurement method at least as accurate as that (relative error less than 20%), we can rule out all other solutions.
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