Mass Luminosity relation doesn't hold true when applied to actual data

[wolf1728]

I see there have been many postings about this topic in this forum.

The formula for this relation is Luminosity = Mass^3.5
Taking logs of both sides we get log (lum) = 3.5 * log (mass)
and using a little algebra we find that the exponent (3.5) should equal log (lum) ÷ log (mass)

I have selected stars from a table located here: http://www.essex1.com/people/speer/main.html
and I have computed what the exponent value should be in each case.
STAR   Lum   Mass   EXP
Orionis C 30,000.00   18.00  3.566653
Becrux   16,000.00   16.00   3.491446
Spica   8,300.00   10.50   3.837759
Achernar   750.00   5.40   3.925568
Rigel   130.00   3.50   3.885439
Sirius A   63.00   2.60   4.336039
Fomalhaut   40.00   2.20   4.678604
Altair   24.00   1.90   4.951367
Polaris A   9.00   1.60   4.674910
Eta Scorpii   6.30   1.50   4.539354
Procyon A   4.00   1.35   4.619371
Alpha Centauri A   1.45   1.08   4.827943
The Sun   1.00   1.00
Mu Cassiopeiae   0.70   0.95   6.953637
Tau Ceti   0.44   0.85   5.051600
Pollux   0.36   0.83   5.483033
Epsilon Eridani   0.28   0.78   5.123395
Alpha Centauri B   0.18   0.68   4.446371
Lalande 21185   0.03   0.33   3.162872
Ross 128   0.0005   0.20   4.722706
Wolf 359   0.0002   0.10   3.698970


No matter whether we concentrate on low mass or high mass stars, we see that the exponent value varies quite a bit .
Does anyone know why there is so much variance in the mass luminosity relation?

 

[Bandersnatch]

A few things you should know about the mass-luminosity formula:

The exponent value is an approximation. It varies from ~2.5 for stars of masses lower than the Sun's through ~4 to ~3 for masses much larger than that. 3.5 is just an average, a useful "rule of thumb".
http://hyperphysics.phy-astr.gsu.edu/hbase/astro/herrus.html#c3
http://www2.astro.psu.edu/users/rbc/a534/lec18.pdf

The reason for the variation is the changing way the energy is transported in the star(i.e., is there convection or not), and the contribution of radiation pressure in mainitaining hydrostatic equilibrium.

There are other factors affecting the exponent value, such as metallicity and age
http://arxiv.org/pdf/1205.5484.pdf

The relation is an empirical one, derived from observations of binary(and multiple) systems, as these are (usually)the only ones for which mass can be determined accurately.

Masses of single main-sequence stars are usually inferred from the relation.

Additionally, the Mass-Luminosity relation concerns only main-sequence stars(stellar clasification includes the Roman "V" number). It is not applicable to e.g., giants, like Beta Crucis, Rigel, or Spica.

Furthermore, the table you're using is very inaccurate. Most of the masses and luminosities do not match other sources I've found, including the arxiv article linked above(table 1). Looking up the sources for mass and luminosity of specific stars on wikipedia should give you more reliable data.
For example, Altair has been measured to have the luminosity of ~10.6 times solar, not 24, and the mass is found to be ~1.79 solar masses, not 1.9
http://iopscience.iop.org/0004-637X/636/2/1087/pdf/0004-637X_636_2_1087.pdf
(also an example of a star whose mass has been found by a different method)Lastly, I recommend this resouces if you want to get yourself acquainted with the guts of stellar astrophysics:
http://www2.astro.psu.edu/users/rbc/teaching.html

Explore subjects in "astro 501" and "534".

 

[wolf1728]

Well, thanks for the quick reply and very informative answer.
I know the Internet is not 100% error-free but you would think that data posted to a scientific web page would be more reliable and accurate than someone's personal Facebook account.
I never knew that a scientific formula could be so approximate as the mass-luminosity relation.
I am currently working on a new web page that will explain the relationship between stellar mass and luminosity, absolute magnitude, temperature, spectral class and longevity.
Thanks again.

 

[SteamKing]

Scientific data is constantly subject to revision. New measurement techniques are being developed which change older data. In astronomical observations of distant objects, a lot of the data may be only an estimate of the true values, since direct measurement is impossible.

 

[Bandersnatch]

@SteamKing: makes you wonder, though, how old is the data if you see Pollux listed with such attributes as in that table. It doesn't even agree with the nice big picture above it.

I know the Internet is not 100% error-free but you would think that data posted to a scientific web page would be more reliable and accurate than someone's personal Facebook account.

I've got nothing against Mr.Speer, but a high school chemitsry teacher's personal webpage hardly qualifies as a scientific source. If only he had included the sources for his table, you'd be able to track it down and see at a glance if it isn't horribly outdated.

The bottom line is, always look for sources.