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## Binary Data Base with Supplementary Mass-Luminosity Relation

The place to discuss creating, porting and modifying Celestia's source code.
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t00fri
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### Binary Data Base with Supplementary Mass-Luminosity Relation

Hi all,

please see here for earlier discussion on this:
viewtopic.php?f=2&t=13885&st=0&sk=t&sd=a&start=15

The requirement for a supplementary Mass-Luminosity relation seems to be rather unavoidable for filling the many empty slots in the binary data base with sensible content!

Here is a general scheme of apparent magnitude assignments for both components of binary star systems that I am currently exploring in an experimental manner:

The formula that 'bdm' already exposed earlier, amounts physicswise to assume that the star luminosities in a multiple system are additive. In astronomy, luminosity is the amount of energy a body radiates per unit time, and energies are additive...

$$\frac{L_{AB}}{L_\bigodot} = \frac{L_A}{L_\bigodot} + \frac{L_B}{L_\bigodot}$$

Here I have conveniently introduced our Sun's luminosity $$L_\bigodot$$ as a reference.

By exploiting, moreover, the approximate equality of distances $$d_{AB}\approx d_A\approx d_B$$ for far away binary systems, this relation can be simply expressed in terms of apparent magnitudes m that are of interest, here:

$$10^{-0.4\,m_{AB}} = 10^{-0.4\,m_A} + 10^{-0.4\,m_B},$$

which is equivalent to the formula bdm used above. For the point of this discussion, let's not worry about how well this formula works in reality. At least, it is based on the sensible assumption that the component luminosities add up to give the system's total luminosity L_AB.

Next I adopt the following strategy:

I compute the apparent magnitude m_A of the A-component from a standard Mass-Luminosity relation that has been extensively tested in case of binary stars with known masses:

$$\frac{L_{A}}{L_\bigodot} \approx \left( \frac{mass_A}{mass_\bigodot}\right )^b.$$ The power b ranges between 3.0 and 4.0, depending on the mass range. b=3.5 (cf figure) is usually a good compromise. For practically all of my binaries, the mass ratios $$\frac{mass}{mass_\bigodot}$$ are known!

The standard relation between luminosity and apparent magnitude for m_A reads:
$$m_A = m_\bigodot -2.5\,\log_{10}\left [\frac{d_\bigodot^2}{d_A^2}\,\frac{L_A}{L_\bigodot}\right ]$$

hence, by inserting the Mass-Luminosity relation, we get:

$$m_A = m_\bigodot -2.5\,\log_{10}\left [\frac{d_\bigodot^2}{d_A^2}\,\left( \frac{mass_A}{mass_\bigodot}\right )^b\right ].$$

To guarantee consistency with the apparent magnitude of the unresolved system m_AB that is always available, I finally compute m_B via the above relation expressing the additivity of luminosities:

$$10^{-0.4\,m_B} = 10^{-0.4\,m_{AB}} - 10^{-0.4\,m_A},$$

or equivalently,

$$m_B = -2.5\,\log_{10}\left(10^{-0.4\,m_{AB}} - 10^{-0.4\,m_A}\right ).$$

In summary: This outlined scheme will assign apparent magnitudes for each of the components that automatically are consistent with the known apparent magnitude of the entire binary system. What will remain is to assert the degree of accuracy of the used Mass-Luminosity relation for the Celestia binaries...

Fridger

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t00fri
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### Re: Binary Data Base with Supplementary Mass-Luminosity Relation

Here is the first result that indicates that a more complex treatment of the Mass-Luminosity relation is unavoidable:

For the 39 spectroscopic binaries from the paper of Pourbaix, I generated the following plot of the mass (in solar units) versus the visual luminosity (in solar units). For ALL binaries In Pourbaix's paper, the visual apparent magnitudes of both components are given! In the figure below, the blue points refer to the A-component and the green dots refer to the B-component. The red line in this double logarithmic plot, corresponds to the expected power behaviour L ~ mass^3.5. Obviously, there is a VERY large scatter of the points, far from what is usually displayed. This is no surprise, since on the y-axis I did not yet plot the bolometric luminosity, but rather the visual one!

What is the bolometric luminosity or magnitude, respectively?

Normally, brightness of a star is measured with instruments that are only sensitive within a small band of wavelength. The Mass-Luminosity relation refers, however, to the star's luminosity (radiation) with ALL wavelengths included! The bolometric correction takes care of this. It's a correction that must be made to the absolute magnitude of an object in order to convert an object's visible magnitude M_vis to its bolometric magnitude M_bol. Mathematically, such a calculation can be expressed as:

M_bol = M_vis + BC(T_eff)

T_eff is the temperature of the surface of the photosphere that give the total luminosity by Planck's famous Blackbody radiation: L = kR^2Teff^4

There are empirical fits of the bolometric correction (BC) as function of the effective temperature T_eff, like this one:

BC = - 8.499*[log(T_eff)- 4]^4 + 13.421*[log(T_eff)- 4]^3 - 8.131*[log(T_eff)- 4]^2 - 3.901*[log(T_eff)- 4] - 0.438

Of course, in Celestia's code, one can also find expressions for the bolometric correction.

So far so good....Let's see what behaviour we get, after the bolometric correction has been included!

Interesting, isn't it?

Fridger
Last edited by t00fri on 06.07.2009, 12:14, edited 1 time in total.

bdm
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### Re: Binary Data Base with Supplementary Mass-Luminosity Relation

t00fri wrote:Interesting, isn't it?
Oh, yes.

ajtribick
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### Re: Binary Data Base with Supplementary Mass-Luminosity Relation

This kind of thing should be very useful for dealing with the MSC. Getting round to writing some code to parse the various barycentre component names in a vaguely sensible way...

Topic author
t00fri
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### Re: Binary Data Base with Supplementary Mass-Luminosity Relation

ajtribick wrote:This kind of thing should be very useful for dealing with the MSC. Getting round to writing some code to parse the various barycentre component names in a vaguely sensible way...

Yeah!...That was a major part of my motivation for returning to this stuff...

Fridger

ajtribick
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### Re: Binary Data Base with Supplementary Mass-Luminosity Relation

My investigation into the designations within MSC reveal a couple of anomalies, sufficiently small numbers to resolve manually

IDS 16471-4140 (V1007 Sco) has two levels (1,11) designated AB: the implied hierarchy is ((Aa-Ab)-B)-B.

Can probably fix by using 1=AC, giving ((Aa-Ab)-B)-C
or using 11=AC, giving ((Aa-Ab)-C)-B

IDS 20146+4025 has two levels (11,111) designated Aab: the implied hierarchy is approximately ((Aa-Ab)-Ab)-B)

Can probably fix using 11=Aac, giving ((Aa-Ab)-Ac)-B
or using 111=Aac, giving ((Aa-Ac)-Ab)-B
or using 111=Aa12, giving ((Aa1-Aa2)-Ab)-B
(following the Washington Multiplicity Catalog convention of using 1,2 to represent the next level of the hierarchy after lower Roman letters)

But I'm going to do some more investigation of these systems to see if I can find solutions to these in the literature.

There are also several instances where the subsystem is designated as, e.g. Ca, implying components C and Ca, however C would then also represent the barycentre, probably can fix these by converting Ca -> Cab, Da -> Dab, etc. Doing so does not cause any duplications of level designations.