A few unanswered questions.

[Apollo7]

Ok this is a question that has plagued me for many many months. Perhaps you guys can put it to rest.

Why, oh why does root(L/Lo)=D.
Where L = Luminosity of Star B
and Lo = Luminosity of the Sun
and D = Distance in AU

Let me elaborate a bit here. Take my favorite star Wolf 359. And yes I realize I posted this before ( I think ) but its been a hell of a long time and the question is still puzzling me, anyway.

Wolf 359 absolute mag = 16.55
so
2.5^(4.85-16.55) = 0.000022, so Wolf 359 visually is only 22/1,000,000 as brilliant as the sun. So, by taking the square root of 0.000022 we get: 0.004699 AU, since we know the absolute magnitude and the distance we can find the apparent magnitude at the said distance, so:

m=16.55-5+5(log(0.004699/206264.81))
and we find that m = -26.662099. So this isn't exactly as bright as the Sun at 1AU (-26.72) but its pretty close, so we see that root(L)=D, in this case, where D = a distance in AU where the comparison star will appear as bright as the sun.

Conversely there is another way to find the D, which is using the equation 10^((Mv-m-5)/-5)=D in parsecs.

We can write this as 10^((16.55+26.72-5)/-5)=D and we multiply D by 206264.81 (1 Parsec in AUs) to find the Distance. This gives us a value for D of 0.004575 AU, a figure that is mostly in agreement with the earlier value of 0.004699 AU. Anyway all this leads into the proverbial question, why in the hell does Root(L)=D?!?!

There must be some reason I'm missing, or something I'm not seeing, though at the moment it just doesn't make much sense to me.

Anyway, any help would be appreciated, I'd like to understand this better.

[revent]

Luminosity and magnitude are not the same thing.

Luminosity is the rate at which an object emits energy.

Magnitude is a logarithmic scale of how bright an object appears (at one parsec, for absolute magnitude).

Not the same thing, not even similar. Luminosity is linear, magnitude is logmarithmic.

For instance, luminosity is a is an absolute characteristic of a star,

Specifically, (m = magnitude, b = brightness, l = luminosity, omega = stefan-boltzman constant) :

M(2) - M(1) = 2.5 [log B(1) - log B(2) ] = 2.5 log [B(1) / B(2)]

L = (surface area of star) x (surface flux) = (4 Pi R^2) x [ OMEGA x (surface temp)^4 ]


If you know the solar constant (earth's flux from the sun), what 1 AU is, the solar radius, and the sun's apparent magnitude you can relate these.

Your similar numbers are just blind luck.

[granthutchison]

I feel I may be missing something about your question, but isn't this just the old inverse square law? In contrast to what revent says, this is just the way the universe works, and it's always going to come out with this sort of relationship.
If a star is intrinsically four times brighter than the Sun, then you need to be twice as far away for it to seem the same brightness in the sky - ie sqrt(4) AU, because apparent brightness falls off with the square of the distance.
Or are you asking something more complicated?

Grant

[Guest]

Revent, luminosity and magnitude are closer to each other than you think.
The absolute visual magnitude (visual magnitude at 10 parsecs) and the visual luminosity measure exactly the same thing, just on different scales - they can be interconverted using a simple log equation. Likewise, the absolute bolometric magnitude and the bolometric luminosity measure the same thing, and can be similarly interconverted.
Apollo7 hasn't specified whether he's using bolometric or visual magnitudes/luminosities, but so long as he doesn't mix bolometric and visual (which he hasn't done, since there's an intact chain of calculation), the relationship he finds is a real one.

Grant

[Apollo7]

well I'm not using anything bolometric, or any real luminosity measurements. "Lv" is a convienant term to designate brightness, which some people DO use to represent luminosity. Typically If I want to find L I'll use the equation L=4piR^2sT^4 where s = the stephan-boltzmann constant. and then I can relate that bit of data to the same values for the sun.

My point was I'm simply trying to get why it is the way it is, as someone who has to teach himself I don't have a linear learning patern in which everything is nice and related to everything else. But I digress, -many- people use brightness as luminosity, it seems so (I would guess) because its alot easier to find out than other ways of doing things. But what I'm getting is its a simple inverse square situation, is this accurate?

[granthutchison]

But what I'm getting is its a simple inverse square situation, is this accurate?

Yes. If you increase the luminosity of your star by a factor of x, then to have it appear with the same apparent magnitude in your planet's sky, you need to move the planet outwards by a factor of sqrt(x).

Grant

[revent]

<sighs> I spent about 15 minutes typing a long reply to this, and then lost it because I got disconnected (invalid session). Maybe in a few days I'll re-explain the difference between luminosity and brightness, or a real astronomer will tackle this.

The brief version is that

root(L/Lo)=D

is wrong, and
root (B/Bo)=D/Do is correct (it's just the inverse square law). If he was correct it would mean that more distant stars were more luminous, which is patently false.

Luminosity (L) is an absolute characteristic of a star, while brightness (B)and magnitude (M or m) depend on distance. L is a measure of power, while B and (M or m) are dimensionless.

Revent, luminosity and magnitude are closer to each other than you think.
The absolute visual magnitude (visual magnitude at 10 parsecs) and

Yeah, saying one parsec was a brain fart. Getting old, I guess.

BTW, his calculations are obviously correct, but that's irrelevant because he uses different versions of the same equation both times (apparently you didn't appreciate my dry wit). His confusion obviously stems from not understanding that luminosity is an intrinsic property of a star, while brightness (and magnitude, the log scale of it) is based on comparison to a reference star.[/quote]

[granthutchison]

If he was correct it would mean that more distant stars were more luminous, which is patently false.

Ah, now I see where you're coming from - but you've missed the essential detail that Apollo7 tells us:

D = a distance in AU where the comparison star will appear as bright as the sun.

In that case his equation "root(L/Lo)=D" is strictly accurate, and fully compatible with the correct meaning of "luminosity"; but I think you've taken the more general, conventional, meaning of "D", yes?

Grant

[revent]

Yea, I was looking at his later calculations where he stated that definition of D that way, but was looking at D in the original equation as just distance (though it would still really be D/Do with either equation, so that you get a dimensionless quantity. Dividing by one is pretty simple, tho ). When stating the original question he said D was just distance in AU.

My main point was really that luminosity and brightness (magnitude is just a measure of relative brightness) are not equivalent terms, and can't be used interchangably. 'Visual Luminosity' is NOT the apparent brightness of a star (see the Guest's post), it is the luminosity of the star in the visual spectrum, and is an intrinsic characteristic of a given star. To use a star's luminosity in the magnitude equation is incorrect, but it's a common mistake because if you are talking about two stars at the same distance (as in the case of absolute magnitude) the ratio of brightnesses is equal to the ratio of luminosities, so in many cases you can disregard the difference. I think it was not understanding that he was implicitly doing this that was causing the confusion.

This was something that confused a lot of art majors (people who thought they were taking astrology to fufill the Gen Ed science course requirement ) back when I was an astronomy TA.

Anyhow, D as he uses it should still really be the RATIO of distances

[granthutchison]

'Visual Luminosity' is NOT the apparent brightness of a star (see the Guest's post), it is the luminosity of the star in the visual spectrum, and is an intrinsic characteristic of a given star.

We still seem to be at some odd cross-purposes, here. I was the "Guest" above (not logged on because of some forum eccentricity, sorry), and I didn't say anything about apparent brightness. I wrote:

The absolute visual magnitude (visual magnitude at 10 parsecs) and the visual luminosity measure exactly the same thing, just on different scales - they can be interconverted using a simple log equation.

(Emphasis added.)
What I'm saying is that absolute magnitudes measure luminosity, while apparent magnitudes measure apparent brightness. When calculating absolute magnitudes by effectively placing all stars at 10 parsecs, you are making a measurement of their intrinsic luminosity (relative to a defined standard based on index stars). For absolute visual magnitude and visual luminosity, the equation for interconversion looks like this:

Mv = 73.5 - 2.5 log (Lv(cd)/1cd)

where the luminosity is in SI units (candelas), and I've been careful to keep my ratio dimensionless before taking the log . If you want to make the calculation using solar luminosities, you just need to know that the Sun has a luminosity of 2.9e27cd. Notice there's no dependence on distance (or any other variable) in that equation - what it's telling us is that absolute visual magnitude, Mv, is just another way of expressing visual luminosity, Lv.
Which is my point. Which I was making specifically in reply to your statement:

Luminosity and magnitude are not the same thing.

because I think the truth or falsity of that statement depends on the kind of magnitude we're discussing - absolute (false) or apparent (true).

But like you, I'm not sure what Apollo7 is getting at when he tries to equate and/or separate "luminosity" and "brightness". The word "brightness" used on its own is as shifty a concept as "magnitude" used without qualification: do we mean intrinsic brightness (equivalent to absolute magnitude, equivalent to luminosity) or apparent brightness (equivalent to apparent magnitude)?

Grant

[revent]

<LOL>

Now that we've been sufficiently anal in defining our terms, I'll agree that we're both right.

Seriously, the distinction between luminosity and magnitude is that luminosity is an absolute, while magnitude (absolute /or/ apparent) is only relative to some other reference star.

If you toss me through a wormhole to the other side of the universe with a ruler, I'll be able to go find some Xenon (or whatever noble gas they use now), determine the duration of a second, measure the speed of light, skip some steps, and (if you grant me some poetic license) measure the luminosity of the local star.

The point is that /any/ atom of Xenon will work. The luminosity is a property of the star, and that would still be it's luminosity if it was the only star in the universe.

If I couldn't compare the local star to a reference star (that I know the magnitude of), directly or by looking its luminosity and magnitude up, I would NEVER be able to determine the magnitude of the local star. Magnitude is not absolute, it is only relative.

I'll say it again...

Luminosity and magnitude are not the same thing.

That is why I was being so picky about ratios....not because log (Watt) is ill-defined, but because all magnitude is is (yeah, bad english ) a convenient way of expressing ratios. Any time you state the magnitude of a star, you're implicitly giving half of the ratio of it's brightness to that of some (any, that's measured) other star.

[granthutchison]

Ah, good, now we're getting down to where you and I have our reality mismatch.

Magnitude is not absolute, it is only relative.

But it's relative to an absolute standard, which has now been shaken loose from the original index stars and defined in SI units - otherwise I never would have been able to give you the formula I did. The reference point of the visual magnitude scale has been fixed at:

apparent magnitude 0 = 2.65e-6 lux

and of the bolometric magnitude scale at:

absolute magnitude 0 = 3.0e28 W

so as soon as you had measured the luminosity of your star, you'd be able to switch to magnitude units, without fretting about index stars. I to have heard the story about the hapless astronomer on the other side of the universe, but it's now outdated. I don't know when this standard was introduced, but the two textbooks I lifted the numbers above from date from the mid 1980's. Does that postdate your time studying astronomy?

Grant