<p>The mass of a white dwarf continues to be that which it was "born" with. It will not change significantly unless it accreted material from a companion.</p>

<p>The radius of a white dwarf is, to first order, given by the "mass-radius relationship" and this relationship does not involve the temperature.</p>

<p>The mass-radius relationship is appropriate for a "cold" star. "Cold" in this context mean that the pressure that supports the white dwarf is only dependent on density, which is the case for the degenerate electrons in the interior, which have kinetic energies much greater than their thermal energy.</p>

<p>However, degenerate electrons also have excellent thermal conductivity, so white dwarf interiors are isothermal. Yet, they have a density gradient - denser in the middle and less dense answer move outwards. At some point close to the surface the electrons are hot enough to stop being degenerate and the gas pressure becomes temperature sensitive.</p>

<p>What this means is that the radius of a hot white dwarf is bigger than that of a cold white dwarf of the same mass. The effect depends on both the mass of the white dwarf (bigger for lower mass white dwarfs with lower surface gravities) and its age (since white dwarfs cool as they get older).</p>

<p>This effect is not negligible and needs to be properly modelled in order to understand the luminosities of white dwarfs.</p>

<p>The plot below is from <a href="https://arxiv.org/abs/1706.05016" rel="nofollow noreferrer">Parsons et al. (2017)</a> and shows (as points) measurements of masses and radii of white dwarfs in eclipsing binary systems. The lines are model curves for white dwarfs with surface temperatures ranging from zero (dashed line) to 60,000 K (appropriate for a very young white dwarf) in steps of 10,000 K. Clearly, there <em>is not</em> a unique mass radius relationship and the radii do apparently depend on temperature (and core composition), as the model curves suggest.</p>

<p><a href="https://i.sstatic.net/P6vOO.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/P6vOO.png" alt="White dwarf mass-radius relation"></a></p>

CC BY-SA 4.0
2019-04-01T03:23:48.820Z
Thank you for your response. So in simple terms the mass-radius relationship R ~ M^1/3 can be used to get a general idea of what the two properties might be but there's no explicit, all-purpose equation as we know that the radius of a hot white dwarf is larger than that of a colder replica.
White Dwarf
CC BY-SA 4.0
2019-04-01T06:14:43.577Z
@WhiteDwarf The $R/R_{\odot}\sim 0.01 (M/M_{\odot})^{-1/3}$ formula would only apply with any accuracy to low mass white dwarfs, and is the (low mass part of the) dashed line in the picture. It is thus a *lower limit*.
ProfRob
CC BY-SA 4.0
2019-04-01T17:10:18.760Z
@WhiteDwarf I would think there is a parametric formula that could be designed using temperature as a third parameter, but I have not seen the curves above written down in that way.
ProfRob
CC BY-SA 4.0
2019-03-31T21:46:51.673Z
2019-03-31T21:46:51.673Z
ProfRob
<p>The mass should be thought of as set by the history of the white dwarf, and it really doesn't change (unless there is mass transfer from a companion, but you know about that).  The radius is then set by the mass, and it really doesn't change either, because the pressure comes from the internal kinetic energy which is extremely large, and is set by the quantum mechanics of the degenerate electrons.  Also, the white dwarf is not really very luminous, such that it will never, in its entire lifetime, emit much energy compared to the vast amount that is already in there, and that means very little change in radius.  So given the mass, there is really only one radius that is appropriate for the white dwarf, which can be found from a formula called the "mass-radius relation" for a white dwarf.</p>

<p>Then the last thing you asked about is the temperature.  This is almost entirely separate from all the other things we've talked about.  It just depends on the age of the white dwarf, as the dwarf cools, so there is no formula that connects temperature to radius or mass (at least not in an important way, there is always   tiny connections because the white dwarf is not fully degenerate).</p>

CC BY-SA 4.0
2019-04-01T03:24:00.413Z
Thank you for your response. This really helped a lot.
White Dwarf
CC BY-SA 4.0
2019-03-31T19:20:58.933Z
2019-03-31T19:20:58.933Z
Ken G
<p>I'm having a difficult time understanding certain behaviors of white dwarfs.</p>

<p>I understand how mass is lost in the red giant to white dwarf transition process. I understand that white dwarfs can accumulate mass from a partner binary or some other source and I'm aware of the 1.4 M☉ Chandrasekhar limit. I get that they're not losing all their mass; in general they're just cooling and in the end there will be a lone black chunk of crystallized material floating forever in the void. I know they are really insanely hot when formed and very cold after an unfathomable amount of time. But is it that same, initial mass, or does it somehow diminish?</p>

<p>But is that initial mass the mass the remnant has after its cooled to a black dwarf? I guess what I'm asking is that <strong>I'm trying to figure out if there is a correlation between temperature, mass and radius.</strong> Some sort of equation?</p>

CC BY-SA 4.0
2019-03-31T21:50:14.353Z
Dwarves are from Middle Earth.
ProfRob
<white-dwarf>
Do white dwarfs lose mass as they fade to black dwarfs? Is there a correlation between temperature, mass, and radius?
White Dwarf
# Do white dwarfs lose mass as they fade to black dwarfs? Is there a correlation between temperature, mass, and radius?

(Asked by: username_0 on 2019-03-31)

I'm having a difficult time understanding certain behaviors of username_0 dwarfs.

I understand how mass is lost in the red giant to username_0 username_0 transition process. I understand that username_0 dwarfs can accumulate mass from a partner binary or some other source and I'm aware of the 1.4 M☉ Chandrasekhar limit. I get that they're not losing all their mass; in general they're just cooling and in the end there will be a lone black chunk of crystallized material floating forever in the void. I know they are really insanely hot when formed and very cold after an unfathomable amount of time. But is it that same, initial mass, or does it somehow diminish?

But is that initial mass the mass the remnant has after its cooled to a black username_0? I guess what I'm asking is that **I'm trying to figure out if there is a correlation between temperature, mass and radius.** Some sort of equation?

**Tags:** white-dwarf

> **Comments:**
> **username_1**: Dwarves are from Middle Earth.  

---

## 2 answers

### Answer by username_1 on 2019-03-31. Score: 2 | Accepted answer

The mass of a username_0 username_0 continues to be that which it was "born" with. It will not change significantly unless it accreted material from a companion.

The radius of a username_0 username_0 is, to first order, given by the "mass-radius relationship" and this relationship does not involve the temperature.

The mass-radius relationship is appropriate for a "cold" star. "Cold" in this context mean that the pressure that supports the username_0 username_0 is only dependent on density, which is the case for the degenerate electrons in the interior, which have kinetic energies much greater than their thermal energy.

However, degenerate electrons also have excellent thermal conductivity, so username_0 username_0 interiors are isothermal. Yet, they have a density gradient - denser in the middle and less dense answer move outwards. At some point close to the surface the electrons are hot enough to stop being degenerate and the gas pressure becomes temperature sensitive.

What this means is that the radius of a hot username_0 username_0 is bigger than that of a cold username_0 username_0 of the same mass. The effect depends on both the mass of the username_0 username_0 (bigger for lower mass username_0 dwarfs with lower surface gravities) and its age (since username_0 dwarfs cool as they get older).

This effect is not negligible and needs to be properly modelled in order to understand the luminosities of username_0 dwarfs.

The plot below is from [Parsons et al. (2017)](https://arxiv.org/abs/1706.05016) and shows (as points) measurements of masses and radii of username_0 dwarfs in eclipsing binary systems. The lines are model curves for username_0 dwarfs with surface temperatures ranging from zero (dashed line) to 60,000 K (appropriate for a very young username_0 username_0) in steps of 10,000 K. Clearly, there *is not* a unique mass radius relationship and the radii do apparently depend on temperature (and core composition), as the model curves suggest.

[![username_0 username_0 mass-radius relation](https://i.sstatic.net/P6vOO.png)](https://i.sstatic.net/P6vOO.png)

> **Comments:**
> **username_0**: Thank you for your response. So in simple terms the mass-radius relationship R ~ M^1/3 can be used to get a general idea of what the two properties might be but there's no explicit, all-purpose equation as we know that the radius of a hot username_0 username_0 is larger than that of a colder replica.  
> **username_1**: @username_0 The $R/R_{\odot}\sim 0.01 (M/M_{\odot})^{-1/3}$ formula would only apply with any accuracy to low mass username_0 dwarfs, and is the (low mass part of the) dashed line in the picture. It is thus a *lower limit*.  
> **username_1**: @username_0 I would think there is a parametric formula that could be designed using temperature as a third parameter, but I have not seen the curves above written down in that way.  

---

### Answer by username_2 on 2019-03-31. Score: 0

The mass should be thought of as set by the history of the username_0 username_0, and it really doesn't change (unless there is mass transfer from a companion, but you know about that). The radius is then set by the mass, and it really doesn't change either, because the pressure comes from the internal kinetic energy which is extremely large, and is set by the quantum mechanics of the degenerate electrons. Also, the username_0 username_0 is not really very luminous, such that it will never, in its entire lifetime, emit much energy compared to the vast amount that is already in there, and that means very little change in radius. So given the mass, there is really only one radius that is appropriate for the username_0 username_0, which can be found from a formula called the "mass-radius relation" for a username_0 username_0.

Then the last thing you asked about is the temperature. This is almost entirely separate from all the other things we've talked about. It just depends on the age of the username_0 username_0, as the username_0 cools, so there is no formula that connects temperature to radius or mass (at least not in an important way, there is always tiny connections because the username_0 username_0 is not fully degenerate).

> **Comments:**
> **username_0**: Thank you for your response. This really helped a lot.  

---

