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EDG
2007-Jul-11, 03:42 AM
I've been looking for a simple formula (or graph) for this but haven't found one anywhere... does anyone know of a formula that relates the temperature and luminosity of a (solo) white dwarf to its age? I know that they get cooler and dimmer as time passes, but I'm not sure of the exact relation.

What I'm after is the luminosity and surface temperature of a 0.593 solar mass white dwarf that became a WD 2.2 billion years ago.

tusenfem
2007-Jul-11, 09:20 AM
I've been looking for a simple formula (or graph) for this but haven't found one anywhere... does anyone know of a formula that relates the temperature and luminosity of a (solo) white dwarf to its age? I know that they get cooler and dimmer as time passes, but I'm not sure of the exact relation.

What I'm after is the luminosity and surface temperature of a 0.593 solar mass white dwarf that became a WD 2.2 billion years ago.

Well, I just took one of my favourite books from my student's time from the shelve: "Black Holes, White Dwarfs and Neutron Stars" by Shapiro and Teucholsky. I think you might want to look at Chapter 4.2, where you will find equation (4.2.8) which gives a cooling time. The equations before that seem to give you the information you need (I think). Let me know if you have the book, otherwise I can send you a copy of the pages involved.

StupendousMan
2007-Jul-11, 01:05 PM
I've been looking for a simple formula (or graph) for this but haven't found one anywhere... does anyone know of a formula that relates the temperature and luminosity of a (solo) white dwarf to its age? I know that they get cooler and dimmer as time passes, but I'm not sure of the exact relation.

What I'm after is the luminosity and surface temperature of a 0.593 solar mass white dwarf that became a WD 2.2 billion years ago.

You could go to the Astrophysics Data Service site

and type "white dwarf cooling time" into the "Abstract words" box, then press the "Submit query" button. You'll see a list of many recent papers on the topic, most of which may be read in full via the links to their astro-ph preprints.

antoniseb
2007-Jul-11, 01:33 PM
Following StupendousMan's suggestion, I found this (http://arxiv.org/PS_cache/astro-ph/pdf/0701/0701738v2.pdf) paper, which shows the white dwarf magnitudes and colors for a particular nearby globular cluster (NGC 6397 - 7200 ly away). While I didn't find a numerical formula in the paper for what you're looking for, there is clearly a simple relationship for white dwarfs from shortly after formation until about ten billion years later. After that it appears that parameters other than age start to become more important. See the graphs starting on page 21 for some insight into the answer to your question.

Spaceman Spiff
2007-Jul-11, 03:12 PM
Have a look at Winget et al. (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1987ApJ...315L..77W&amp;data_type=PDF_H IGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf) ApJ Letters, 315, L77, 1987:

Eyeballing a graph of their Table 1, a 0.6 Msun carbon white dwarf cooling for 2.2 billion years should have a luminosity of about 2E-4 that of our Sun.

You can also have a look at another excellent paper here (http://www.astro.caltech.edu/%7Esrk/ay125/Hansen2004.pdf).

EDG
2007-Jul-11, 03:20 PM
Thanks all... I'll see if I can trawl through those papers later on (and tusenfem - no, I don't have that book). Do those graphs/papers tell you how hot the dwarf is too, or just the luminosity?

antoniseb
2007-Jul-11, 03:29 PM
You might be able to work out how hot the surface is by knowing the surface area and total luminosity. Also, the paper I referred to includes color as the horizontal axis, and you could potentially derive temperature from that.

Spaceman Spiff
2007-Jul-11, 04:02 PM
RE. Effective surface temperature vs. time - see Figure 23 of the second paper I linked to. Caution - the surface temperature of a white dwarf is always much cooler than its nearly isothermal interior (where the electron degeneracy equation of state predominates). Also, as mentioned just above, you can use the blackbody radiation formula:

L = 4*pi * R^2 * sigma_sb * T_eff^4
or
L/Lsun = (R/Rsun)^2 * (T_eff/5777 K)^4

to figure out the effective temperature for a given luminosity L and radius R. The radius of a "cold, ideal" 0.6 solar mass (carbon) WD is about 0.0125 solar radii. For the relative luminosity of 2e-4 and relative radius of 0.0125, I find T_eff ~ 6145K - which matches up well to the detailed theoretical prediction in Figure 23 of that second paper I mentioned.

David Sims
2013-Oct-17, 08:51 PM
There aren't any helium-core white dwarfs yet because stars having insufficient mass to fuse helium into heavier elements have not yet had enough time to leave the main sequence. All white dwarfs currently existing are oxygen-carbon-neon white dwarfs.

For these:
log t = 6.7 − (5/7) log L

Where t is the white dwarf's age in years, L is the luminosity in units of the sun's luminosity (3.846e26 watts). Log is base 10, of course.

Here are some curvefits, so you don't have to do the whole model thing.

0.25 < M/M๏ ≤ 0.45,
R/R๏ = 0.07279307 (M/M๏)² − 0.0752974 (M/M๏) + 0.03327478

0.45 < M/M๏ ≤ 1.2,
R/R๏ = −0.010421 (M/M๏) + 0.018821

1.20 < M/M๏ ≤ 1.41
R/R๏ = −0.0814246 (M/M๏)² + 0.1899852 (M/M๏) − 0.1044496

David Sims
2013-Oct-17, 09:07 PM
So Spaceman Spiff's value for the luminosity of a 2.2 billion year old white dwarf is about right. The curvefit I gave you gives about the same thing, L/L๏ = 1.998e-4. The radius of a 0.6 solar mass carbon-oxygen core white dwarf is, according to the other curvefit, R/R๏ = 0.0125684, again about what Spiff said. The temperature from the Stefan-Boltzmann law is 6134K.

Ken G
2013-Oct-17, 11:59 PM
There aren't any helium-core white dwarfs yet because stars having insufficient mass to fuse helium into heavier elements have not yet had enough time to leave the main sequence.Sure there are, see http://online.kitp.ucsb.edu/online/snovae-c07/marsh/pdf/Marsh_SnovaeConf_KITP.pdf.
A helium white dwarf is made by a special binary channel, which in essence "speeds up" their formation time by allowing them to start out more massive and get a head start on their evolution.
All white dwarfs currently existing are oxygen-carbon-neon white dwarfs.
Blanket generalizations are hard to get away with in astronomy, it has so many tricky ways to do things we don't expect. See for example http://arxiv.org/abs/astro-ph/9911371 for ideas on how iron-rich white dwarfs might be produced.

PetTastic
2013-Oct-18, 08:03 AM
When calculating the surface area of a white dwarf, how much extra do you add for its atmosphere?
I am seeing numbers that assume 1e-3 to 1e-7 solar masses of atmosphere, as this is not degenerate matter it must have a significant volume compared to the white dwarf core.

Ken G
2013-Oct-18, 12:48 PM
When calculating the surface area of a white dwarf, how much extra do you add for its atmosphere?Nothing significant. The atmosphere is very thin, it's scale height is kT/mg, where T is perhaps twice the Sun's, but g is perhaps 10 times the Sun's. The Sun's atmospheric scale height is few hundred km, so a white dwarf atmospheric scale height is less than one hundred km, but it's radius is about 5,000 km. That's less than a 2% effect.