Luminosity - Mass Relation from 6th Catalog of Binary Orbits

[t00fri]

Hi all,

after our long and interesting discussions about how to extract and convert /many/ orbits for binary stars from catalogs into Celestia's new "multiple star facility", here is a /very brief/ status report of what I have been trying most recently:

During the last two days, I did quite a few studies of how one might arrive at sensible estimates for the masses of both stars separately for visual binaries in the 6th catalog of orbits...

One ansatz would use theoretical calculations for star evolution, as suggested by Dr. Evil. I studied the respective original papers in detail. The underlying physics I can understand very well. I think there is lots of information we may exploit soon or later...

Another promising ansatz was to explore observations for binary stars directly.

In both cases things boil down to the empirical fact that notably main sequence stars tend to exhibit a clearcut correlation between their total luminosity and their mass and also between their color (temperature) and their mass. So knowing the luminosity or the color temperature would provide some kind of info about the masses!

I was just curious to learn how far one can get here...

As a start I used the quoted visual magnitudes, Kepler's 3rd law along with the tabulated orbit elements and the /distances/ from the Hipparcos catalog, to derive with a Perl script plots of the

total luminosity versus the total mass

of these binary systems.

Kepler's 3rd law gives the total mass (m1+m2) in terms of the semi-major axis and the orbit's period. The total luminosity I obtained from adding the individual luminosities and combining apparent magnitude and distance information. What is probably missing is a bolometric correction... but I'll get at that...

In a first illustration below, I display the result from the 6th catalog for all star orbits with quality grade better than 3 on a scale 1 ...5.

The luminosity in units of the solar luminosity is plotted versus the total system mass (m1 +m2) in units of the solar mass. The pink dot corresponds to our sun, of course.

If there is interest, I can detail all my calculations, of course, but Dr. Evil usually steps on my feet if I add a single formula to my posts



Of course, there are many things to add, here. Let me know whether you are curious...

Bye Fridger

[maxim]

One ansatz would use theoretical calculations for star evolution, as suggested by Dr. Evil. I studied the respective original papers in detail. The underlying physics I can understand very well. I think there is lots of information we may exploit soon or later...

Ansatz = approach
Dr. Evil = Evil Dr.

sorry to be pert. I couldn't resist

maxim

[Evil Dr Ganymede]

Dr. Evil = Evil Dr.

Thanks for pointing that out, Maxim.

First. I am not "Dr Evil", it's "Evil Dr Ganymede".

Second. I do not "step on your feet if you add a single formula to your posts". All I did was say that you usually dive straight into mathematical or physical explanation with plenty of jargon and little qualitative description, which makes it hard for others to follow what you say.

Fridger - I could help out here since I know a fair bit about stellar evolution. But I am really not inclined to offer that help so long as you continue to antagonise me in the way you have been doing. It's up to you.

[t00fri]

One ansatz would use theoretical calculations for star evolution, as suggested by Dr. Evil. I studied the respective original papers in detail. The underlying physics I can understand very well. I think there is lots of information we may exploit soon or later...

Ansatz = approach
Dr. Evil = Evil Dr.

sorry to be pert. I couldn't resist

maxim

Sorry for insisting:

1) Ansatz = Ansatz!

In science this is a word that has been exported from German, since there is no precise counterpart for it in English. Believe me. I just KNOW that.


2) Dr. Evil = Dr. Evil,

when I have an argument with him with no hope for peace...and it's

Evil. Dr.

when we are peacefully and creatively talking (asto)physics which can also happen . If you have a look in my posts you will clearly spot the differences...

Bye Fridger

[maxim]

In science this is a word that has been exported from German, since there is no precise counterpart for it in English. Believe me. I just KNOW that.

I believe you. It's just new to me. What exactly is the different meaning? To my opinion 'approch' fits very well.

maxim

[granthutchison]

In science this is a word that has been exported from German, since there is no precise counterpart for it in English. Believe me. I just KNOW that.

I believe you. It's just new to me. What exactly is the different meaning? To my opinion 'approch' fits very well.

Ansatz has a more specific meaning in scientific English than I think it does in German: it involves using a particular set of assumptions in order to solve a problem. The best English equivalent might be to say that an ansatz is a "way of getting a handle on something", if you know that expression.

Grant

[t00fri]

In science this is a word that has been exported from German, since there is no precise counterpart for it in English. Believe me. I just KNOW that.

I believe you. It's just new to me. What exactly is the different meaning? To my opinion 'approch' fits very well.

maxim

The point is that an ansatz is a specific kind of approach, yet not every approach consitutes an ansatz! A rigorous approach, for example, in the mathematical sense, definitely does NOT represent an ansatz.

The german translation for "approach" is : "Zugang" (to a problem). So an approach is a fairly general term.

An "ansatz" is an approach with the understanding that the initial assumptions cannot be proven, but nevertheless appear sensible.

I hope this was reasonably clear.

Bye Fridger

[maxim]

Thanks Grant, Fridger,

I think I got it. 'Heuristic approach' would certainly not hit it, but point in the direction.

maxim

[Cham]

To me, Ansatz = "guess" or "hypothesis".

In physics, we talk about an Ansatz solution as a good guess solution to some difficult problem, which after calculations reveal itself as a right solution. I never understood why exactly we never used the word "guess", in place of Ansatz. Maybe because there's a bad perception of that word, and "Ansatz" feels better.

[t00fri]

To me, Ansatz = "guess" or "hypothesis".

In physics, we talk about an Ansatz solution as a good guess solution to some difficult problem, which after calculations reveal itself as a right solution. I never understood why exactly we never used the word "guess", in place of Ansatz. Maybe because there's a bad perception of that word, and "Ansatz" feels better.

Guess is too weak. "Educated guess" might be better. Ansatz is more than guessing and less than rigorous, a very delicate balance. It so happened that the German word "Ansatz" is used frequently in scientific English.
I can't help it

Bye Fridger

[t00fri]

Hi all,

while people seem much more interested in the origin of the word "Ansatz" than in my displayed results above, I nevertheless have quickly "LaTeX'ed " what precisely went in. I then converted the standard PS output into a JPEG. Writing math with the forum's editor is a little tedious. LaTeX I can do as fast as I can type...

Bye Fridger

[t00fri]

Hi all,

the next steps might be as follows.

After contemplating about possible bolometric corrections (there seems to be a smallish systematic shift relative to Sol in my plot above), there are two options at hand:

1) One studies the extensive papers and calculations by the Univ. Geneva/ch group about Luminosity vs. Mass predictions on the basis of stellar evolution theory. The underlying physics is fairly transparent in principle, while the resulting calculations are complicated. Vast lookup tables exist as pointed out by Evil Dr. Ganymede, previously.

What mainly enters is

--the choice of an equation of state ,
--opacity tables, treatment of convection atmosphere and mass loss
--all the nuclear processes that take place in stars
--input about the initial abundances of elements

In case of stars heavier than Sol, a simple equation of state suffices, while for the lighter stars things become more evolved.

Output is then the thermodynamics of the stars, notably the effective temperature and luminosities...

In short, one may explicitly compare the above observational data from the binary systems with those theoretical calculations and examine various hypotheses of universality of luminosity <-> mass or (color) temperature <-> mass correlations. Such investigations would clearly provide a feel on how big the uncertainties would typically be.

2) A much more simple and more empirical approach would be to just run off a standard "least-square" fit program with a well-guessed trial function, depending on some parameters to render it flexible. The program would then just find the best approximation through my above points.

By assuming that the functional dependence luminosity(mass) is about universal, the mass e.g. of the primary/seconary can be estimated from its luminosity.

One will certainly get out some numbers. The crucial question will be whether we can believe them and what the uncertainties will be...That's what requires different approaches to gain more confidence.

Bye Fridger

[wcomer]

Fridger,

The Keplerian approach for calculating the mass is fine as long as the distance error is small. If it is unknown then you need to use the mass-temp relationship as that is just one equation one unknown (mass). Luminosity-mass is one equation with two unknowns (distance and mass) so I don't see how it helps unless you assume you already had a good estimate for distance. Using your estimate of total mass (dependent on distance) doesn't help as it is still one equation two unknowns (distance and total mass.) If you use the ratio of the masses (independent of distance) then you could have a system of three equations (mass-luminosity for both masses and mass ratio) for three unknowns (the two masses and the distance.) However that last approach seems messy, requires that the mass-luminosity relationship not be a perfect power law in order for the system to be solvable, and most likely gives lousy results.

BTW, the chart you have shown would be much cleaner if you compared mass ratios with luminosity ratios as neither ratio has a distance dependency and there is no built in non-linearity. Or to be a bit more verbose, the binary masses ratio can be easily measured from relative maximum angular separation between each star and the barycenter. Even without that data, the ratio should be easy to estimate from the orbital elements and presumably this estimate will not have any distance dependency. Finally, comparing sums of masses with sums of luminosities introduces a non linearity which isn't present when comparing their ratios.

cheers,
Walton

[t00fri]

Hi Walton,

many thanks for your comments.

Fridger,
The Keplerian approach for calculating the mass is fine as long as the distance error is small. If it is unknown then you need to use the mass-temp relationship as that is just one equation one unknown (mass). Luminosity-mass is one equation with two unknowns (distance and mass) so I don't see how it helps unless you assume you already had a good estimate for distance. Using your estimate of total mass (dependent on distance) doesn't help as it is still one equation two unknowns (distance and total mass.)

There must be a misunderstanding here: I only have one unknown, namely (m1+m2), for which I have solved.

The 6th catalog of binary orbits provides for each star:

-- the Hipparcos catalog number
-- the apparent visual magnitudes both for the primary & secondary star
-- the period of the orbit [years]
-- the semi-major axis of the orbit [arcsecs]

Since the Hipparcos catalog number is given, I then look up the parallax i.e. the distance in the Hipparcos catalog via Perl. So I have used two catalogs for the above plot.

Since most of the binary star orbits of grade 1,2 refer to relatively nearby systems, the parallax is well measured and so is the distance.

If you use the ratio of the masses (independent of distance), then you could have a system of three equations (mass-luminosity for both masses and mass ratio) for three unknowns (the two masses and the distance.) However that last approach seems messy, requires that the mass-luminosity relationship not be a perfect power law in order for the system to be solvable, and most likely gives lousy results.

As to my investigations, I actually began with a consideration of the temperature|color <-> mass correlations. In my opinion, the best strategy is anyhow to combine all methods available. This will help tremendously to reduce the systematic errors involved.

The above plot is only a small part of what I looked at during the past week
(where I had vacations ).

I just thought that illustrations based on the Keplerian masses might be easiest to understand for non-experts. An explicit discussion of the results from stellar evolution theory requires much more know how in (astro)physics.

Let me summarize again my underlying "working hypothesis" at this point:

-- I assume that there is a universal relation of (total) luminosity <-> mass:

Code: Select all

            L=  L(mass)


-- in case of two nearby masses m1+m2 (i.e. small orbit parallax), it makes sense to me to add the corresponding luminosities.

Note that the luminosity (including bolometric corrections) measures the total luminous energy radiated into space.

Indeed, it seems that for the data from the 6th catalog of binary orbits, this procedure reproduces essentially the universal L(mass) relationship we know from other sources! That's essentially what I have tested so far with the above plot.

-- Once that universal functional form L(mass) has been determined either from (binary) data (see my plot + fits) or from stellar evolution theory, the idea is to apply it to the individual components as well (thereby exploiting the universality assumption). After all, the 6th catalog also provides the individual luminosities! So I would just "slide a bit" on the universal L(mass) curve to read off m1, say, given the value of L1.

In view of the small parallax of the system, inherent nonlinearities should be small...

--What is certainly correct is that the Keplerian mass determination depends on the 3rd power of the distance, hence this induces substantial uncertainties, if the latter is poorly known. However, also the "classical" method that you mentioned ( observing the movement of the barycenter relative to fixed background stars) requires both long observation times and relatively nearby stars!

Bye Fridger

[t00fri]

Could anyone point me to a lookup table where I can convert the detailed color +subcolor classes of the Hipparcos catalog into temperature via Perl??

Thanks
Fridger


Of course, I am aware of the main color class - temperature relation. The subclasses are the point.

[granthutchison]

Could anyone point me to a lookup table where I can convert the detailed color +subcolor classes of the Hipparcos catalog into temperature via Perl??

Celestia's own look-up table is here, where you'll also find details of the bolometric conversions. I took the latter from Lang's Astrophysical Data: Planets and Stars, which also provides quite detailed lists of temperature by spectral class and subclass for luminosity classes I, III and V. (I haven't compared Celestia's temperature conversion to Lang's in detail, but the data seem to be similar.)

Grant

[chris]

Could anyone point me to a lookup table where I can convert the detailed color +subcolor classes of the Hipparcos catalog into temperature via Perl??

Celestia's own look-up table is here, where you'll also find details of the bolometric conversions. I took the latter from Lang's Astrophysical Data: Planets and Stars, which also provides quite detailed lists of temperature by spectral class and subclass for luminosity classes I, III and V. (I haven't compared Celestia's temperature conversion to Lang's in detail, but the data seem to be similar.)

The temperature data comes right from Lang's. However, one thing I haven't yet done is create different tables for luminosity classes I, III, and V. Currently, all stars are using the spectral type to temperature tables for class V. I suppose that now would be a good time to fix this . . .

I'm very excited about this work on incorporating a catalog of star orbits into Celestia. Does any other astronomy program account for orbital motion when plotting stars? It strikes me that this might be a unique feature of Celestia, but I don't really know.

--Chris

[chris]

I just checked in new temperature tables for giant and supergiant stars. There are also new tables for Wolf-Rayet stars, though I don't imagine any of the systems in the catalog of star orbits contain a Wolf-Rayet member.

--Chris

[granthutchison]

The temperature data comes right from Lang's.

Ah. The code annotation reads "Star temperature data from Carroll and Ostlie's _Modern Astrophysics_ (1996), p. A-13 - A-18."

However, one thing I haven't yet done is create different tables for luminosity classes I, III, and V.

OK, that explains my vague feeling that there was some mild mismatch between Lang and Celestia's data.

Grant