What is Nilssen's Law of Stars and how does it relate to star data?

[Widdekind]

By inspection of the following star data (Solar units), we find that:

M ~ R^3/2 (Nilssen's Law)​

From this we can derive further scaling laws, for average density (p) and surface gravity (g):

p ~ R^-3/2
g ~ R^-1/2
p ~ g³​

Surface gravity (g) acts as a "compressive force" on the star's gases, rapidly increasing its average density (p).

Star-Type   Mass   Radius    Predicted-Mass     Density
O           64.0     16.0           64.0          0.016
B           18.0      7.0           18.5          0.052
A            3.1      2.1            3.0          0.33
F            1.7      1.4            1.7          0.62
G            1.1      1.1            1.0          0.83
K            0.8      0.9            0.85         1.1
M            0.4      0.5            0.35         3.2

http://en.wikipedia.org/wiki/Stellar_classification​

 

[Vanadium 50]

I'm afraid a) this is nothing new, b) not quite right, and c) of limited applicability. Taking them in reverse order, this is true for main sequence stars only, and there is a clear break in the law around one solar mass: below it, the exponent is around 1.7 or 1.8 (not your 1.5) and above it it's closer to 1.2. This has to do with the onset of convection. Finally, this has been known for a very long time - it goes back at least to the 1970's and quite probably the 1930's.

 

[Widdekind]

I listed above the Predicted Masses (R^1.5) for all (Main Sequence) star types, and the agreement is substantial for the lot of them. Therefore, a constant exponent of 1.5 is applicable for all of them, assuming the masses & radii I cited are accurate.

Do you dispute those Wikipedia statistics ?

Are you referring to computer models instead ?

 

[Vanadium 50]

Do you dispute those Wikipedia statistics ?

I tried to be polite. I did. Honest.

Actually, I dispute both your data collection procedure which stopped at Wikipedia (and ignored the obvious point of the sun, at a radius and luminosity of the sun). Had you used more data points - and the data are out there - you would have seen that there is clearly a break in the curve.

I dispute your ability to do curve fitting, as a simple best fit returns a smaller exponent.

I also am not impressed with your scholarship, as the idea of a power-law relationship or relationships between radius and mass is many decades old.

 

[Widdekind]

Cite another set of statistics. If Wikipedia is wrong, sign up & correct it.

Cite authors who have acknowledged power-law relations between Radius & Mass. To my knowledge, such laws are not described in Carroll & Ostlie (Intro to Mod. Astr.), or Bowers & Deeming (Stars).

As it is, I have cited sources, and you have not.

 

[Vanadium 50]

Cite another set of statistics. If Wikipedia is wrong, sign up & correct it.

Good heavens. There are thousands of stars in dozens of catalogs. Wikipedia picked seven numbers to be more or less representative.

Cite authors who have acknowledged power-law relations between Radius & Mass.

This is a solution of the Lane-Emden equation. Emden published it in 1907. This is not new.

 

[Vanadium 50]

Furthermore, if you use the larger and more complete table at Wikipedia's entry on http://en.wikipedia.org/wiki/Main_sequence" , you would find a best fit exponent of 1.3, not 1.5. You would also see that the power law doesn't fit well to the curve: the exponent changes over the course of the curve.

A best fit of 1.3 corresponds to an average polytrope of n = 1.9. This is sensible - for a completely non-convective star you expect n = 3 and for a completely convective one you expect n = 1.5.

 

[Widdekind]

The mass of a Lane-Emden "star" is:

$$M_{LE} = \int 4 \pi \rho r^{2} dr$$
$$= 4 \pi \rho_{c} S^{3} \int \theta^{n} x^{2} dx$$​

where we have factored out the Scale Height:

$$S = \sqrt{\frac{(n+1) P_{c}}{4 \pi G \rho_{c}^{2}}}$$​

But:

$$P_{c} = K \rho_{c}^{1+\frac{1}{n}}$$ (Polytropic Equation of State)​

And:

$$P_{c} = \frac{k_{B}}{m} \rho_{c} T_{c}$$ (Ideal Gas Law)​

So:

$$K = \frac{k_{B}}{m} \frac{T_{c}}{\rho_{c}^{\frac{1}{n}}}$$​

And:

$$S = \sqrt{\frac{(n+1) k_{B}}{4 \pi G m} \frac{T_{c}}{\rho_{c}}}$$​

Therefore:

$$M_{LE} = 4 \pi \left\{ \frac{(n+1) k_{B}}{4 \pi G m} \right\}^{\frac{3}{2}} \frac{T_{c}^{\frac{3}{2}}}{\rho_{c}} \Gamma_{n}$$​

where:

$$\Gamma_{n} = \int \theta^{n} x^{2} dx$$​

is a dimensionless integral whose value depends on the Polytropic Index n. Thus,

$$M_{LE} \approx \frac{T_{c}^{\frac{3}{2}}}{\rho_{c}}$$​

This only accords w/ Nilssen's Law if:

$$T_{c} \approx \rho_{c}^{-1/3}}$$​

CONCLUSION: Nilssen's Law is a more restrictive claim than Lane-Emden alone, including an additional Temperature constraint.


REFERENCES:

• http://en.wikipedia.org/wiki/Lane-Emden_equation
• http://mathworld.wolfram.com/Lane-EmdenDifferentialEquation.html

 

[Vanadium 50]

Yesterday you were unaware that such a model even existed:

Cite authors who have acknowledged power-law relations between Radius & Mass.

So I am not impressed by posting a bunch of formulas today.

My objections stand:

(1) This is not new. After chasing around, I've found that these kinds of relationships were being discussed in 1907.

(2) This is not correct: using the Wikipedia table I posted, one gets an exponent of 1.3, not 1.5 and it's clear that the curve does not fit a simple power law.

(3) This is of limited applicability: it's true for main sequence stars only, even though your derivation assumes nothing about the star being on the main sequence.

I also took a look at the Wikipedia article where you say you got your table from, and was amazed to find something different than what you posted there. There are ranges for masses and radii, not single numbers. One cannot pick a number from the range to match one's theories.