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I have a question regarding the White Dwarf radius formula given on wikipedia, in terms of what units I am supposed to use and what expected values of one variable would be.

https://en.wikipedia.org/wiki/White_dwarf#Mass%E2%80%93radius_relationship

Listed here.

I think I have everything working, and used the right units I think (I used the eV-s version of the reduced plank constant, kg for electron mass, and of course kg-m-s2 for gravitational constant, kg for mass of the star), but I am not sure how to get the elctrons per unit mass.

So, for instance, I have a hypothetical star of 0.4 solar masses, just for a way to test the formula. But I can't get the right answer for the formula unless I know how many electrons per unit mass there are. So how exactly is this decided upon? What is a reasonable range for this value?

Like, for instance, I put in a thousand and get a radius of 1515331.7009 metres or 1515.3317 kilometres or 0.0021 solar radii. Which I know is too small, as lower mass White Dwarves should be more massive.

47.5047 seems to line up with the Radius approximation of 0.01(M)^-1/3. So would that actually be closer to the right answer? Edit: I got units wrong there, it is actually closer to 2998.8042. Not 47.5047. My mistake.

Just, please help me with finding what the values should be for electrons per unit mass. I am most confused by this condundrum.

DanceroftheStars
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  • See https://astronomy.stackexchange.com/questions/22494/question-about-number-density-of-electrons For many small nuclei there is 1 electron for two nucleons eg Carbon 12 has 6 electrons. – James K Dec 04 '23 at 00:07
  • @JamesK That doesn't seem what it is asking for here. it is asking for electrons per unit mass, not electrons per nucleus. And putting in two for that just gets out a white dwarf of only 48 metres in radius, which is definitely very, very off from the reality. It most definitely a much higher number. But it also does not seem like moles are an accurate measure either as using anything to do with moles makes the radius way too high. So it is somewhere between those, which does not narrow it down much, sadly. – DanceroftheStars Dec 04 '23 at 04:17
  • What do you mean with 'eV-version of the planck constant'? You have to use the same units for everything. $h=6.62607015 × 10^{-34}m^2 kg / s$. You cannot add 12 apples and 1kg of oranges to get 13 kg of fruit salad. This said, 0.5 is a good first guess for electrons per unit mass – planetmaker Dec 04 '23 at 07:37
  • And please always give the units for your numbers. It helps you as much as the reader. 47000 is no measure of size, nor is 115000. However 47000km might be. If you plug in numbers into an equation, plug in the units, too. If you don't get the expected units, check your inputs and/or conversions – planetmaker Dec 04 '23 at 08:13
  • Checkout reference 40 cited just above equation for units. N is $\beta/m_p$ with $\beta$ about 0.5. – eshaya Dec 04 '23 at 14:03
  • @planetmaker I tried that before, but it just makes the numbers way too small when I use m2 kg/s. It is also the reduced planck constant, not the planck constant. Putting in 0.5 electrons per unit mass and that value of the Reduced Planck Constant (I divied by 2pi to make it the Reduced form), I get a radius of 3.1035025054×10−39 metres. – DanceroftheStars Dec 04 '23 at 14:42
  • I can only get reasonable numbers, on the scale of thousands of km, when I make the number density in the thousands with the previous units for \hbar , or octillions when I use the one you suggested. Which is confusing if 0.5 is supposed to be accurate, which the reference does indeed state. – DanceroftheStars Dec 04 '23 at 14:45
  • You cannot just plug-in numbers until you get something which roughly matches your expectation... Anyway, following the reference of the cited equation in wikipedia, it seems that the quote is incomplete and misses a factor of $m_p^{5/3}$ in the denominator where $m_p$ is the proton mass. That gives me diameters of the order of 1000km. I'd still expect a factor of 10 more... – planetmaker Dec 04 '23 at 15:16
  • $N = 0.5$, $hbar = 1.05457e-34 Js$, $me = 9.109e-31 kg$, $G = 6.674e-11 $ N/m²/kg² , $mp = 1.672e-27 kg $

    print( N(5/3) * hbar2 / ( 2 * me * mp(5/3) * G * (Msun)(1/3) ) )

    972780.113378007

     – planetmaker Dec 04 '23 at 15:22
  • Okay, thanks, I added the proton mass and now have a 1316.808 km radius. That is still overly low, but is far closer, thanks for pointing that out. Though it overall seems that the other formula given on the page is far more reasonable, with it giving me 27144.1761 km radius. Which is actually in stellar ranges, around 0.0389 Solar Radii. Which seems reasonable for a White Dwarf. Also, I found that the two equations change at the same rate, or at least very close to the same rate. – DanceroftheStars Dec 04 '23 at 17:16
  • The numbers of electrons per unit mass would be a number, per unit mass. i.e. The number of electrons per kg. – ProfRob Dec 04 '23 at 19:06
  • @ProfRob I feel I solved the issue, the linked page says that the average number for that is 0.5 for the average White Dwarf. And the approximations it gets are rather close to the actual radius when I plug actual White Dwarfs into the equation. For the simplified form of the equation, that is, the original form is still off by a large factor. But thanks for your help, I think I am satisfied with the way that the simplified form is working, It isn't perfect of course, but it is satisfactory for me outside of extremes such as ZTF J1901+1458. – DanceroftheStars Dec 04 '23 at 19:10
  • Right, so it is the reciprocal of $\mu_e$, the number of mass units per electron. – ProfRob Dec 04 '23 at 19:52

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