White Dwarf Cooling Model: Core and Atmosphere Temperature Relationship

[Chronothread]

Perhaps this should be on the homework forum but I'm not sure, so I put it here.

How do you calculate the cooling time scale of the surface of a white dwarf and the surface of the core of a white dwarf? I have an equation for the cooling scale of a white dwarf in general but I'm not sure how exactly to correlate this to the cooling scale of the surface and the cooling scale of the surface of the core. Even just a suggestion on how to relate those two would be great, but something more in detail is always appreciated.

Thanks for your time.

 

[mgb_phys]

You would need the thermal conductivity of the star's atmosphere
I'm not sure if there is a standard model of this but you could assume the atmosphere was all Helium and lookup the optical crosssection.

 

[Orion1]

atmospheric parameters of cool white dwarfs...

Bergeron, Ruiz, and Leggett, for example, estimate that after a carbon white dwarf of 0.59 solar mass with a hydrogen atmosphere has cooled to a surface temperature of 7,140 K, taking approximately 1.5 billion years, cooling approximately 500 more kelvins to 6,590 K takes around 0.3 billion years, but the next two steps of around 500 kelvins (to 6,030 K and 5,550 K) take first 0.4 and then 1.1 billion years.

Reference 2 has 'atmospheric parameters of cool white dwarfs' - Table 2 pg. 36

Reference:
http://en.wikipedia.org/wiki/White_dwarf#Radiation_and_cooling"
http://www.journals.uchicago.edu/doi/pdf/10.1086/312955"

 

[Chronos]

It's a very difficult question with no easy answer. A white dwarf is not powered by fusion, rather by exotic processes that are not very well understood. Absent destabilizing influences, like a binary partner feeding it fuel [see supernova], a white dwarf can endure for many, many billions of years.

 

[Orion1]

white dwarf thermal distribution...


I plotted all of the data from three different parameters from reference 1, Table 2 - pg. 36 as attachment.

The thermal distribution is dynamic, with only a small thermal distribution distinction between a Hydrogen and a Helium atmosphere composition. The Hydrogen atmospheres corresponding with lower mass, temperature and age. The Helium atmospheres corresponding to higher mass, temperature and age.

First attachment:
Left: 3D plot, x axis: solar masses, y axis: temperature (K), z axis: age (Gy) in rainbow spectrum.
Right: 2D plot, x axis: solar masses, y axis: temperature (K), color: atmosphere, red: Hydrogen, blue: Helium.

Mathematica 6 source code:

ListPointPlot3D[{{ .79, 6270, 5.07},{ .57, 8200, 1.4},{ .74, 6550, 3.1},{ .71, 6020, 3.74},{ .62, 4780, 7.48},{ .51, 5360, 3.58},{ .84, 6750, 4.66},{ .58, 5200, 4.67},{ .58, 5320, 4.25},{ .58, 4830, 6.01},{ .59, 5550, 3.39},{ .82, 9810, 1.45},{ .57, 7260, 1.91},{ .25, 7140, .61},{ .35, 7720, .71},{ .61, 7320, 1.46},{ .56, 4530, 7.58},{ .66, 5490, 4.81},{ .71, 6820, 2.4},{ .73, 5190, 6.77},{ .57, 6680, 1.69},{ .33, 5220, 1.93},{ .68, 6860, 2.69},{ .6, 5490, 3.85},{ .58, 5220, 4.61},{ .58, 5140, 4.91},{ .57, 6150, 2.81},{ .68, 5620, 4.59},{ .64, 6650, 2.74},{ .59, 6450, 1.95},{ .37, 5280, 2.08},{ .59, 6590, 1.85},{ .8, 6430, 4.84},{ .78, 5080, 8.09},{ .72, 5790, 4.53},{ .76, 5940, 4.51},{ 1, 7270, 3.74},{ .58, 4370, 7.4},{ .6, 4930, 5.96},{ .57, 6580, 2.42},{ .6, 7280, 1.44},{ .68, 7030, 2.51},{ .27, 4900, 2},{ .56, 4760, 6.53},{ .39, 4170, 4.72},{ .6, 4870, 6.85},{ .72, 5740, 4.83},{ .58, 11940, .52},{ .59, 6100, 2.23},{ 1.11, 10390, 2.84},{ .76, 7190, 3.11},{ .58, 10240, .79},{ .35, 8340, .59},{ .59, 6030, 2.29},{ .88, 8780, 2.34},{ .59, 6950, 1.63},{ .58, 5350, 4.13},{ .59, 7160, 1.51},{ .55, 7330, 1.78},{ .63, 4640, 7.29},{ .59, 6430, 1.97},{ .6, 9680, .7},{ .57, 5810, 3.3},{ .57, 4780, 6.58},{ 1.2, 4490, 6.47},{ .59, 5770, 2.69},{ .58, 4910, 5.73},{ .59, 5910, 2.42},{ .57, 7710, 1.63},{ .33, 4000, 4.12},{ .69, 5600, 4.86},{ .64, 4170, 8.78},{ .57, 5540, 3.9},{ .6, 7450, 1.36},{ .56, 5050, 4.79},{ .58, 8690, 1.21},{ .59, 6180, 2.17},{ .49, 4690, 4.96},{ .57, 4970, 5.75},{ .58, 5030, 5.28},{ .83, 4990, 8.52},{ .58, 4830, 5.99},{ .87, 6810, 3.82},{ .59, 6040, 3.07},{ .6, 7450, 1.36},{ .58, 5520, 3.49},{ .58, 10680, .71},{ .74, 6520, 3.15},{ .82, 5640, 6.72},{ .27, 6340, .89},{ .76, 6470, 4.26},{ .78, 4810, 8.97},{ .69, 6490, 3.35},{ .26, 5200, 1.58},{ .63, 4630, 8.33},{ .53, 6900, 1.42},{ .73, 5840, 4.52},{ .6, 7640, 1.28},{ .59, 7140, 1.52},{ .6, 9150, .81},{ .33, 5520, 1.55},{ .82, 4590, 9.68},{ .58, 12230, .48},{ .86, 6880, 4.57},{ .58, 10170, .81},{ .57, 5490, 4.03},{ .49, 5810, 2.03},{ .58, 8290, 1.37}}, ColorFunction->"Rainbow"]

Reference:
http://www.journals.uchicago.edu/doi/pdf/10.1086/312955"

 

[petrichor]

Just assume it is a blackbody cooling by radiative energy loss. You certainly don't need the thermal conductivity of the atmosphere (wtf?) and the energy generated is pretty much nil compared to the leftover heat from the original stellar collapse.

 

[Orion1]

Newton's law of cooling...


Do white dwarfs obey Newton's law of cooling?

Newton's law of cooling differential boundary conditions solution:
$$T(t) = T_{e} + (T(0) - T_{e}) e^{-r t}$$

Environmental temperature equal to Universe temperature (cosmic microwave background temperature):
$$\boxed{T_e = T_u = 2.725 \; \text{K}}$$

Solving for the time constant:
$$\boxed{r = \frac{1}{t} \ln \left( \frac{T_u - T(0)}{T_u - T(t)} \right)}$$

Bergeron, Ruiz, and Leggett model (post# 3):
Time constant for a white dwarf cooling from 7140 K to 6590 K in 0.3 billion years:
$$\boxed{r = \frac{1}{dt} = 8.470 \cdot 10^{-18} \; \text{s}^{-1}}$$

Differential solution based upon model:
$$\boxed{\frac{dr}{dt} = \frac{1}{dt^2} = -1.070 \cdot 10^{-34} \; \text{s}^{-2}}$$

Mathematica 6 best fit for differential solution based upon model:
$$\boxed{\frac{dr}{dt} = \frac{1}{dt^2} = -9.764 \cdot 10^{-35} \; \text{s}^{-2}}$$

White dwarf core surface environmental temperature equal to atmosphere temperature:
$$\boxed{T_{e} = T_{a}}$$

White dwarf core surface temperature:
$$\boxed{T_{c}(t) = T_{a}(t) + (T_{c}(0) - T_{a}(t)) e^{-r t}}$$

Solving for the time constant:
$$\boxed{r = \frac{1}{t} \ln \left( \frac{T_{a}(t) - T_{c}(0)}{T_{a}(t) - T_{c}(t)} \right)}$$

Thermal conductivity constant:
$$\boxed{k_T = \frac{dE}{dL \cdot dt \cdot dT} = \frac{\text{W}}{\text{m} \cdot \text{K}}}$$

White dwarf atmosphere thermal conduction:
$$\boxed{\frac{dQ}{dt} = 4 \pi k_T \left( \frac{r_c^2}{dr_a} \right) [T_{c}(t) - T_{a}(t)]}$$

$$r_c$$ - core radius
$$dr_a$$ - atmosphere shell thickness

Thermal conductivity constants:
Hydrogen: 180.5
Helium: 0.1513
Diamond: 900 - 2320

a white dwarf with surface temperature between 8,000 K and 16,000 K will have a core temperature between approximately 5,000,000 K and 20,000,000 K. The white dwarf is kept from cooling very quickly only by its outer layers' opacity to radiation.

Reference:
http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation"
http://en.wikipedia.org/wiki/Heat_transfer#Newton.27s_law_of_cooling"
http://en.wikipedia.org/wiki/Thermal_conductivity"
http://en.wikipedia.org/wiki/Hydrogen"
http://en.wikipedia.org/wiki/White_dwarf#Atmosphere_and_spectra"

 

[Orion1]

white dwarf atmosphere albedo...



White dwarf core surface temperature:
$$\boxed{T_{c}(t) = \frac{T_{a}(t)}{(1 - \alpha_{a})^{\frac{1}{4}}} \sqrt{ 2 \left( \frac{r_c + dr_a}{r_c} \right)}}$$

Solving for white dwarf atmosphere reflectivity (albedo):
$$\boxed{\alpha_a = 1 - 4 \left( \frac{r_c + dr_a}{r_c} \right)^2 \left( \frac{T_a(t)}{T_c(t)} \right)^4}$$

White dwarf model relationship between time constant and reflectivity:
$$\boxed{T_{c}(t) = T_{a}(t) + (T_{c}(0) - T_{a}(t)) e^{-r t} = \frac{T_{a}(t)}{(1 - \alpha_{a})^{\frac{1}{4}}} \sqrt{ 2 \left( \frac{r_c + dr_a}{r_c} \right)}}$$

Solving for atmosphere temperature:
$$\boxed{T_a(t) = \frac{\sqrt[4]{1 - \alpha _a} T_c(0)}{e^{r t} \left( \sqrt{2 \left( \frac{r_c + dr_a}{r_c} \right)} - \sqrt[4]{1 - \alpha_a} \right) + \sqrt[4]{1 - \alpha_a}}}$$

Solving for original core surface temperature:
$$\boxed{T_c(0) = \left(e^{r t} \left(\frac{\sqrt{ 2 \left( \frac{r_c + dr_a}{r_c} \right)}}{\sqrt[4]{1 - \alpha_a}} - 1 \right) + 1 \right) T_a(t)}$$

$$r_c$$ - core radius
$$dr_a$$ - atmosphere shell thickness
$$\alpha_{a}$$ - white dwarf atmosphere reflectivity (albedo)

Reference:
http://en.wikipedia.org/wiki/Black_body#Temperature_relation_between_a_planet_and_its_star"