# Thread: Luminosity and apparent magnitude calculations

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## Luminosity and apparent magnitude calculations

Hello everyone,
I've been reading up on apparent magnitude and luminosity, and have tried using some of the formulas that I've read about to calculate some of the apparent magnitudes of common stars, and it doesn't seem to be working too well.

So, according to Wikipedia,
$m_{x}= -2.5\log_{10}(F_x/F_x^0)$
where Fx is the observed flux in the passband x, and F0x is a reference flux.

In turn, flux can be calculated as:
$F=\frac{L}{4\pi r^2}$
where L is the luminosity and r is the distance between the source and the observer.

So, I tried calculating the flux of Vega, for reference, using the values for luminosity and distance found on Wikipedia (I converted to Watts and meters):

$F= \frac{1.42043\times10^{28}\;W}{4\pi(2.39351369\times10^{17}\;m)^2}$

$F=1.97305\times10^{-8}\;W/m^2$

and then calculated the flux of a test star, in this case Fomalhaut:

$F= \frac{6.779673\times10^{27}\;W}{4\pi(2.3651321\times10^{17}\;m)^2}$

$F=9.64469\times10^{-9}\;W/m^2$

But then when I put it into the formula:

$m= -2.5\log_{10}(\frac{9.64469\times10^{-9}\;W/m^2}{1.97305\times10^{-8}\;W/m^2})$

I get:

$m= 0.777124544902120$

But, Fomalhaut's apparent magnitude is supposed to be 1.16.
I'm obviously doing something wrong, but I can't figure out what.

Any help would be much appreciated!

2. Your formula requires that the reference flux correspond to a visual magnitude of zero. Is that true for Vega? If not, your formula is for the difference in magnitude between Vega and Fomalhaut.

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Yes, I realize that, and yes Vega's magnitude is roughly 0.03, and while this is not a true zero-point, it was good enough to be used by astronomers.
So, since Vega's magnitude is 0.03, then 1.16 - 0.03 = 1.13, but the function didn't give anywhere close to 1.13.

I think it might have something to do with the fact that the luminosities that are given on Wikipedia are of all wavelengths, which is obviously a problem. X-rays don't contribute much to visual brightness.

But I can't find a source of data that breaks down a star's luminosity into specific wavelengths or bands. But perhaps there's another way to calculate that? Perhaps by using the temperature and treating the star as a black body?

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A couple of things:

* can you give the source for your data on the two stars? I suspect that Wikipedia may not be giving you reliable info

* While Vega has, pretty much by definition, a magnitude of 0, you have to be very careful to specify the passband (or waveband or filter or ...).

Stars' magnitudes are often given without the crucial qualify of what passband is implied or used; this can make a huge difference! And of course if the star emits most of its energy (luminosity, flux) in a passband other than the 'visual', all bets are off.

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Nereid:
The data that I used was from Wikipedia. As you said, it may not be reliable.
http://en.wikipedia.org/wiki/Vega
http://en.wikipedia.org/wiki/Fomalhaut

The luminosity is given in units of Solar Luminosity, which I converted to Watts by multiplying by 3.839 x 10^26. I also converted the distances from light years to meters.
And I agree, the passband is probably affecting the results dramatically. I'm guessing that the luminosities that Wikipedia gives, if they are correct, represent the total luminosity at all wavelengths, which is not very useful.

But as I said, I haven't found a place to find luminosity separated out into wavelengths or passbands. I'm still wondering if there's a way to calculate this using the star's temperature and radius (which are also available on Wikipedia... haha).

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Originally Posted by TetsudaiIchiban

But as I said, I haven't found a place to find luminosity separated out into wavelengths or passbands. I'm still wondering if there's a way to calculate this using the star's temperature and radius (which are also available on Wikipedia... haha).
You are right about the flux being wavelength dependent; here's some values you can use for the bands. ...the variation may be significant in the infrared.

http://www.astro.ljmu.ac.uk/~ikb/con...its/node1.html

G^2

7. Fomalhaut and Vega are both A stars, so neither will have significant emission in the IR (I don't believe their dust disks are significant to the total flux), and both spectra will be pretty similar-- Vega is a little hotter, but that would mean its flux would be enhanced by UV emission that doesn't count toward apparent magnitude, a correction that would make the problem even worse, not better. Your numbers seem to be what Wiki is using-- you can do it straight from the luminosities given (37 for Vega, 17.66 for Fomalhaut, and note there is no need to convert into any other unit because all you care about is the ratio anyway). Those numbers do give a magnitude difference of 0.8, not 1.15. I can only conclude that the Wiki has an inconsistency. It is generally held that Fomalhaut is more than one magnitude less bright to the eye, so the problem could be with the luminosity of Vega. I've heard larger numbers for that, like around 50 solar, not 37.

8. Interesting. Fomalhaut and Vega are right about the same distance from the earth (25 ly), just over 90 degrees apart in the sky, and part of the same moving group. They have to be 35 ly apart.

ETA: looking it up in SkyMap, it appears that Vega luminosity should be 47 not 37 solar, about what Ken was thinking.

9. I found on another website a luminosity for Vega of 52 L sun and Fomalhaut of 16.5 suns. This will give closer answer to the accepted value.
for Vega
http://www2.potsdam.edu/islamma/Phys335Ch08StarProp.htm
for Fomalhaut
http://onlinelibrary.wiley.com/doi/1...10339/abstract
I get 1.24 after the difference of the magnitude of Vega is taken into account.
Plus if you put all the errors in your calculations (I forget how) the accepted value would probably be within the range of values.
Last edited by TheOncomingStorm; 2010-Oct-20 at 09:15 PM. Reason: subtracter instead of added

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Aha! Brilliant! So it seems that the zero-reference (Vega) was wrong. Strange, I wonder why Wikipedia lists the luminosity as such.

Aha! This site: http://www.solstation.com/stars/vega.htm explains the discrepancy.
It states that Vega's luminosity is 37 +/- 3 times the Sun's up "to 58 (pole on) times its luminosity." Now it makes sense! Because of Vega's being oblate due to its rapid rotation, its apparent luminosity changes depending on the point of view. I suppose then that it doesn't act like a point source, which is why different luminosities are listed.

And unless I'm mistaken, the line of sight from the Earth is at most only a few degrees off from the polar axis of Vega, so it's luminosity as observed from the Earth would be closer to the "pole on" value (> 50) rather than from the side.

And as it turns out, when you try using 52 as its luminosity, it acts as a great zero-point. I tried it with Fomalhaut using Wiki's information, and it gave a magnitude of 1.14! (So, technically, the true apparent magnitude would be 1.14 + 0.03 = 1.17) Furthermore I tried testing with Polaris as well, and got a magnitude of 2.08, not too far off from the magnitude listed in Wikipedia of 1.97.

I think I've learned to be more wary of Wikipedia's information... but it won't stop me from using it. Haha!

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Originally Posted by Davidlpf
I found on another website a luminosity for Vega of 52 L sun and Fomalhaut of 16.5 suns. This will give closer answer to the accepted value.
for Vega
.
Yes, as you and Grapes have said, that is apparently where the discrepancy lies in the Luminosity determination of Vega.
(Flux variation in the optical bands probably wouldn't amount to more than a fraction of 1 percent).

It may seem hard to believe there is such a large discrepancy in the luminosity determination of Vega especially since it has such a high parallax.
However, it is understandable in light of recent discovery of Vega's large rotation rate and the polar / equatorial temperature difference which causes a luminosity difference between poles and equator.
Apparently the equatorial Luminosity is about 37 L sun, and the polar Luminosity is > 56 L sun.
See here:
http://lanl.arxiv.org/abs/astro-ph/0603327
So I suspect that is unfortunate considering we have designated it as the zero point flux.

G^2
Last edited by Gsquare; 2010-Oct-21 at 01:56 AM.

12. Originally Posted by Gsquare
So I suspect that is unfortunate considering we have designated it as the zero point flux.
Purely from an observational point of view, why does it matter? Vega is probably better calibrated against laboratory standards than any other star, well observed across the spectrum, and has no history of significant variability. The only other feature I might ask for is something lying closer to the equator so everyone on the planet could use it directly. As a calibration object, anisotropy of its light output will matter only when our relative motions have changed our viewing angle relative to Vega's pole, by which time the mutual distance will be significantly different as well. (And as some have pointed out, Vega isn't exactly the zero flux point, although it is the most fundamental calibration object for some photometric systems).

Now, if it should start varying, that would be a problem.

13. You are right that the problem with using it as a flux calibrater has nothing to do with flux calibration itself (as you point out), and there is no problem if one follows the appropriate chain from apparent magnitude, to absolute magnitude, to luminosity. But Gsquare has a point that the tendency is to look for the shortcut that leads to luminosity directly from Vega's luminosity (as the OP did). Vega only works as a calibration star if we never use its luminosity (which we shouldn't, but the temptation is there). The Wiki that gives its luminosity should have a caution that it is both uncertain and not relevant to calibration of the luminosity of other stars.

14. This just had me thinking on the drive to work - even nearly circular but inclined planetary orbits around early-type stars could have interesting seasonal effects due to the star's oblateness. Rotational velocities indicate that very oblate stars are very common for spectral types earlier than somewhere around the A-F spectral-type boundary, and Doppler studies of "retired" A stars headed up the giant branch show a significant number of these have planets.

15. That's an interesting point, you are saying that in situations where we can directly detect the brightness of the planet, it may show seasonal variations. I have several questions:
1) how often do giant planets have inclined orbits?
2) what is a "retired" A star?
3) is it important that we look at the planet after the star heads up the giant branch (say to get the luminosity increase)? (Giant stars will tend to slow their rotation and oblateness to conserve angular momentum.)

16. My initial thought was that in an inclined orbit, you would receive more radiation when farthest from the star's equator (twice per orbit).

There are giant planets in inclined orbits - I don't know how large the sample size is by now from measuring the Rossiter-McLaughlin effect (change in star's net Doppler shift as part is occulted during planetary transit). A retired A star is one that has finished its main-sequence lifetime. As such stars climb the red-giant branch, Doppler searches become much more sensitive, because the intrinsically broad hydrogen lines of A stars severely limit the correlation accuracy for radial velocities. As it expands and cools, the spectrum becomes rich in narrow absorption lines, increasing the detectabulity of planets. At this point, folks can find planets around stars in this mass range when they are giants but not dwarfs.

17. Originally Posted by ngc3314
My initial thought was that in an inclined orbit, you would receive more radiation when farthest from the star's equator (twice per orbit).
Oh I see, you are talking about life on the (presumably terrestrial) planet, not observing the planet from Earth.
As it expands and cools, the spectrum becomes rich in narrow absorption lines, increasing the detectabulity of planets. At this point, folks can find planets around stars in this mass range when they are giants but not dwarfs.
OK, but that is probably also because the lines get narrower, but that will also imply the star is becoming less oblate. So the effect you are interested in may go away by the time we can observe it, but that doesn't mean we can't infer it used to be there before we could observe it. Probably the observation would be of more asterobiological interest if we thought A stars could incubate life in the first place, but who knows, maybe they can-- and if so, there might be some quite important seasonal effects for inclined planets. Certainly grist for the sci fi mill, at least!

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Sorry to interrupt the train of thought, but I just realized something. Luminosity is a measure of the total electromagnetic energy that is radiated by a body, so in the case of oblate stars like Vega, what are the mentioned luminosities (i.e. 37 L and 56 L) really defining? Would it be akin to something like an apparent luminosity that would be derived from the greater or lesser flux at a given point of view (due to the oblateness)? So then, regardless of the degree of oblateness, a star should still have a single true luminosity that would be calculated with the oblateness taken into account (as opposed to normal luminosity calculations, which approximate the energy from a black body curve using the temperature and radius, meaning that the assumed shape is a sphere), although this luminosity would likely not be very useful in determining magnitude. Yes?

19. Originally Posted by TetsudaiIchiban
Luminosity is a measure of the total electromagnetic energy that is radiated by a body, so in the case of oblate stars like Vega, what are the mentioned luminosities (i.e. 37 L and 56 L) really defining? Would it be akin to something like an apparent luminosity that would be derived from the greater or lesser flux at a given point of view (due to the oblateness)?
We can think of one version of L as nothing but another way to say F, just in units of L = F*4pi*D2 for known D, and where F is the flux we measure from our angle of perspective on the star. Only then is L useful in calibrating from star to star, but it isn't really its "luminosity". The actual meaning of luminosity, call it Lreal, is the rate the star emits energy into all directions in total, so has to be corrected for oblateness. That's isn't useful going from star to star, because different stars have different oblatenesses. Given that we get the right magnitude of Fomalhaut using an L of 56 or so, that must be the L you'd get if the star emitted the flux we see isotropically, whereas the Lreal of 37 or so reflects the effort to correct down to the real L, given that we see Vega from nearly pole on where it looks extra bright.

So then, regardless of the degree of oblateness, a star should still have a single true luminosity that would be calculated with the oblateness taken into account (as opposed to normal luminosity calculations, which approximate the energy from a black body curve using the temperature and radius, meaning that the assumed shape is a sphere), although this luminosity would likely not be very useful in determining magnitude. Yes?
Exactly.

By the way, this all means the Wiki is accurate, though it could have included a caution how not to misinterpret the L. Another feather for Wiki's cap, it's right up to date.

20. Originally Posted by TetsudaiIchiban
Luminosity is a measure of the total electromagnetic energy that is radiated by a body....
Astronomers call the "TOTAL electromagnetic energy radiated by a body" by a special name: "bolometric luminosity". It is, unfortunately, impossible to measure, since we do not have devices which absorb and record radiation from across the entire electromagnetic spectrum. Astronomers can measure the radiation at a few selected wavelengths, and then guess what the overall spectrum of the object might be, and then integrate over all wavelengths for that estimated spectrum; but that involves a lot of guessing.

So, the "bolometric luminosity" of an object is always only a rough guess.