<p>The <a href="https://en.wikipedia.org/wiki/Vogt%E2%80%93Russell_theorem" rel="nofollow noreferrer">Vogt-Russell</a> &quot;theorem&quot; says that the structure of a star is uniquely determined by its mass and the distribution of chemical elements within its interior.</p>
<p>To answer your question, you need to decide what you are holding fixed. A star of a given mass and composition has a fixed radius. If you increase the mass you increase the radius. If you fix the mass and make to star more metal-poor, then you decrease the radius (but not by very much for M-dwarfs).</p>
<p>The surface temperature is determined by the luminosity and radius. The luminosity is governed by the central temperature. An increase in mass, increases the central temperature and greatly increases the luminosity. The radius also increases, but not by enough to compensate, so the surface temperature also rises. At a fixed mass, a decrease in metallicity does not greatly affect the central temperature or luminosity, so a decreasing radius means that the surface temperature increases.</p>
<p>The basic answer is that the equations of stellar structure determine the structure of the star and there is a limit to how much handwaving you can do.</p>
<p>In recent years it has been realised that the Vogt-Russell theorem is not the end of the story, especially for M-dwarfs. Their radii and surface temperatures also appear to depend on how fast-rotating and magnetically active they are. This is due either to suppression of convective heat flux by interior magnetic fields or the blocking of flux by starspots at the surface (e.g. <a href="https://arxiv.org/abs/1802.04288" rel="nofollow noreferrer">Jackson et al. 2018</a> - work I am involved in). Both of these effects make the stars bigger and cooler at a fixed mass.</p>

CC BY-SA 4.0
2021-02-23T18:21:18.357Z
Okay, I'm mostly looking at a fixed radius.  So, how could I interpret a 0.25 solar radii, 0.25 solar mass M-dwarf such that the radius remains 0.25 solar radii even with a larger mass?  Is that possible in any way?
Xi-K
CC BY-SA 4.0
2021-02-23T19:15:37.240Z
@Xi-K That isn't what your question asks.
ProfRob
CC BY-SA 4.0
2021-02-23T21:21:34.493Z
I'll just move this addendum to another question thread.  Thanks for your other answer.
Xi-K
CC BY-SA 4.0
2021-02-23T10:56:59.427Z
2021-02-23T10:56:59.427Z
ProfRob
<p>For <a href="https://en.wikipedia.org/wiki/Main_sequence" rel="nofollow noreferrer">main sequence stars</a> there is not much choice. If you pick mass, then all the rest is determined; any variation you see is due to different chemistry (thus fractions of different elements).</p>
<p>For the main sequence one generally <a href="http://articles.adsabs.harvard.edu//full/1991Ap%26SS.181..313D/0000319.000.html" rel="nofollow noreferrer">assumes</a> that <span class="math-container">$L \propto M^{3.5}$</span> and <span class="math-container">$R \propto M^{2/3}$</span> (and similarily the density is given for a given mass). The <a href="https://en.wikipedia.org/wiki/Hertzsprung%E2%80%93Russell_diagram" rel="nofollow noreferrer">Hertzsprung-Russel diagramme</a> is expression of this.</p>
<p>Any stars not on the main sequence, be that the white dwarfs or any giants, are there due to their peculiar chemistry. White dwarfs are degenerate stellar remanents, slowly cooling. The giants have <a href="https://en.wikipedia.org/wiki/Stellar_evolution#Main_sequence" rel="nofollow noreferrer">evolved</a> away from main sequence. They have some other fusion process(es) than hydrogen burning in their core so that their luminosity is much bigger than hydrogen-burning stars; at least they have a hydrogen-depeleted and helium-enriched core and started hydrogen-shell burning.</p>

CC BY-SA 4.0
2021-02-23T08:51:09.293Z
2021-02-23T08:51:09.293Z
planetmaker
<p>Assuming a relatively even proportion of mass and radius, a 0.25 solar mass and radius star would have a density of 22.5003 g/cm³, or about 16 times our Sun's density.</p>
<p>However, there is obviously a range of densities that this sized red dwarf could have.  One of the primary factors is its temperature, with cooler meaning a higher density.</p>
<p>My question is three-pronged: what would increase or decrease this sized star's temperature, how would this impact its density, and what factors other than temperature would increase or decrease this star's density?</p>

<star><temperature><size><density><red-dwarf>
What factors influence a star's temperature and density?
Xi-K
# What factors influence a star's temperature and density?

(Asked by: username_0 on 2021-02-23)

Assuming a relatively even proportion of mass and radius, a 0.25 solar mass and radius star would have a density of 22.5003 g/cm³, or about 16 times our Sun's density.

However, there is obviously a range of densities that this sized red dwarf could have. One of the primary factors is its temperature, with cooler meaning a higher density.

My question is three-pronged: what would increase or decrease this sized star's temperature, how would this impact its density, and what factors other than temperature would increase or decrease this star's density?

**Tags:** star, temperature, size, density, red-dwarf


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## 2 answers

### Answer by username_1 on 2021-02-23. Score: 0 | Accepted answer

The [Vogt-Russell](https://en.wikipedia.org/wiki/Vogt%E2%80%93Russell_theorem) "theorem" says that the structure of a star is uniquely determined by its mass and the distribution of chemical elements within its interior.

To answer your question, you need to decide what you are holding fixed. A star of a given mass and composition has a fixed radius. If you increase the mass you increase the radius. If you fix the mass and make to star more metal-poor, then you decrease the radius (but not by very much for M-dwarfs).

The surface temperature is determined by the luminosity and radius. The luminosity is governed by the central temperature. An increase in mass, increases the central temperature and greatly increases the luminosity. The radius also increases, but not by enough to compensate, so the surface temperature also rises. At a fixed mass, a decrease in metallicity does not greatly affect the central temperature or luminosity, so a decreasing radius means that the surface temperature increases.

The basic answer is that the equations of stellar structure determine the structure of the star and there is a limit to how much handwaving you can do.

In recent years it has been realised that the Vogt-Russell theorem is not the end of the story, especially for M-dwarfs. Their radii and surface temperatures also appear to depend on how fast-rotating and magnetically active they are. This is due either to suppression of convective heat flux by interior magnetic fields or the blocking of flux by starspots at the surface (e.g. [Jackson et al. 2018](https://arxiv.org/abs/1802.04288) - work I am involved in). Both of these effects make the stars bigger and cooler at a fixed mass.

> **Comments:**
> **username_0**: Okay, I'm mostly looking at a fixed radius.  So, how could I interpret a 0.25 solar radii, 0.25 solar mass M-dwarf such that the radius remains 0.25 solar radii even with a larger mass?  Is that possible in any way?  
> **username_1**: @username_0 That isn't what your question asks.  
> **username_0**: I'll just move this addendum to another question thread.  Thanks for your other answer.  

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### Answer by username_2 on 2021-02-23. Score: 0

For [main sequence stars](https://en.wikipedia.org/wiki/Main_sequence) there is not much choice. If you pick mass, then all the rest is determined; any variation you see is due to different chemistry (thus fractions of different elements).

For the main sequence one generally [assumes](http://articles.adsabs.harvard.edu//full/1991Ap%26SS.181..313D/0000319.000.html) that $L \propto M^{3.5}$ and $R \propto M^{2/3}$ (and similarily the density is given for a given mass). The [Hertzsprung-Russel diagramme](https://en.wikipedia.org/wiki/Hertzsprung%E2%80%93Russell_diagram) is expression of this.

Any stars not on the main sequence, be that the white dwarfs or any giants, are there due to their peculiar chemistry. White dwarfs are degenerate stellar remanents, slowly cooling. The giants have [evolved](https://en.wikipedia.org/wiki/Stellar_evolution#Main_sequence) away from main sequence. They have some other fusion process(es) than hydrogen burning in their core so that their luminosity is much bigger than hydrogen-burning stars; at least they have a hydrogen-depeleted and helium-enriched core and started hydrogen-shell burning.


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