What is the Anomaly with Sirius B's Data in Astrophysics Calculations?

[Fallguy]

Hi all. New guy here. I signed up for this forum for the specific purpose of asking this, since I can't reconcile the data myself. (I've been trying for days without success)

First of all, a little about me. I am extremely familiar with astrophysics and physics in general. I have no difficulty following a detailed mathematical model. That said, I've been studying the star Sirius B (the White Dwarf companion of Sirius A) and did a few calculations based on the data I could find. Here's the problem...

Input Data: (References are http://arxiv.org/PS_cache/astro-ph/pdf/0306/0306604.pdf" )

Mass (Sol=1): 0.952-1.113
Radius (Sol=1): 0.008004-0.008713
Surface Temp (°K): 25,193
Distance (arcsec; parsecs): 0.38002; 2.631440451
Absolute Magnitude (M): 11.34
Apparent Magnitude (m): 8.44

Now, armed with the above data, we can do a blackbody emission analysis that should show that all these data points agree, right? Wrong. Here's what I got...

Stefan-Boltzmann calculation: (Bolometric Flux w/m²=seT⁴
25,193⁴ = 4.02828E+17 x 5.6705119E-08 x 0.9994 = 2.28286988E+10 w/m²
6,009,189 m radius = 4.53776090E+14 m² x 2.28286988E+10 w/m² = 1.03591177E+25 watts / 3.845E+26 watts (Sol) = 0.02694 Bolometric Luminosity
4.85 - ( Log (0.02694) x 2.5 ) = 8.77 Absolute Magnitude (M)
M + ( 5 x Log (2.631440451 / 10) ) = 5.87 Apparent Magnitude (m)

Compare that to the above RECONS data... Houston, we have a problem. The only way to get the RECONS data with the established radius, temperature, and distance is to use an emissivity of only 0.094... which would indicate that White Dwarfs are not even close to being blackbody radiators. That can't be right...

The only other option is to assume that the RECONS data is not Bolometric Magnitude, but Absolute Visual Magnitude, which is not what it says it is. If we assume that, then we get close with the known data points of radius, temperature, and distance. Visual flux at 25,193°K is only 8.22% of the total output (based on Planck Radiation Density l_IE = ((2phc²)/l⁵)/(e^(hc/lkT)-1) for frequencies 380nm-760nm using 10nm frequency steps) From that we get...

Absolute Visual Magnitude (Mv): 11.49
Apparent Visual Magnitude (mv): 8.59

...which is close to the RECONS data, but still off by a significant margin.

Double-checking the math, I input Sol and Sirius A, both of which match the RECONS data within a very close order of magnitude. Here's the RECONS data for those two stars.

Sol: M = 4.85; m = -26.72
Sirius A: M = 1.47; m = -1.43

Sirius A mandates that you assume an effective emissivity of only 0.89, (which could very well be the case with all the dust in the Sirius system clouding the images we get, making both values in error) or you have to assume that the 9,900°K T_eff is wrong and use 9,630°K. (which is much closer to other A0V star temperatures) Sol comes out right on the money. (big surprise )

If RECONS is using Mv and mv for its data, then Sol and Sirius A are wrong. (Sol Mv = 5.70; mv = -25.87 / Sirius A Mv = 2.40; mv = -0.49 by computation) If RECONS is using M and m for its data, than Sirius B is wrong.

So... am I missing something here? Did I do my math right? Did I forget a step that can reconcile this data? I'll admit my own fallibility, but I think I got it all right. Any help on this is greatly appreciated.

 

[SpaceTiger]

The only other option is to assume that the RECONS data is not Bolometric Magnitude, but Absolute Visual Magnitude, which is not what it says it is.

It looks to me that both the table headers and column descriptions list the magnitudes as "V" and "Mv", which are certainly visual magnitudes. More precisely, they're probably Johnson V band magnitudes, so in order to compare them to the fluxes expected from a blackbody curve, you'd have to integrate over both the blackbody curve and the filter throughput function. If you don't want to bother with that, there might be some standard bolometric corrections tabulated in the literature, though I'm not immediately sure where.

 

[Chronos]

SpaceTiger is correct, the catalog entries are not bolometric, they are Johnson band magnitudes. To convert absolute magnitude [Mv] to bolometric magnitude, you need to know the bolometric correction [BC] factor - which is approximately 1.01 for Sirius B [solar BC is about -0.07]. The conversion formula is pretty simple:
M(bol) = Mv + BC.

 

[Fallguy]

It looks to me that both the table headers and column descriptions list the magnitudes as "V" and "Mv", which are certainly visual magnitudes. More precisely, they're probably Johnson V band magnitudes, so in order to compare them to the fluxes expected from a blackbody curve, you'd have to integrate over both the blackbody curve and the filter throughput function. If you don't want to bother with that, there might be some standard bolometric corrections tabulated in the literature, though I'm not immediately sure where.

Ok, so I found the proper spectral filter for Johnson V band http://www.stsci.edu:8082/hst/acs/documents/handbooks/cycle14/c10_ImagingReference8.html". No joy.

Maybe I'm not understanding what the Filter function is. From what I can see, Johnson V-band cuts all frequencies below 460 nm and above 620 nm completely out of the equasion. But it also reduces the energy density of all the frequencies between the two points to 37.4% to 3.7% of normal. Variable depending on wavelength, peak at 550 nm)

I modeled the Johnson V-band curve exactly, accurate to within a fraction of a percent by 10 nm steps. Here's the raw energy density for Sol at those frequencies and the energy density after applying the filter.

.   [FONT="Symbol"]l[/FONT]          ((2[FONT="Symbol"]p[/FONT]hc[sup]2[/sup])/[FONT="Symbol"]l[/FONT][sup]5[/sup])/(e[sup](hc/[FONT="Symbol"]l[/FONT]kT)[/sup]-1)         E[FONT="Symbol"]l[/FONT][sub]I[/sub]          V[sub]J[/sub]       E[FONT="Symbol"]l[/FONT][sub]I[/sub]V[sub]J[/sub]
0.00000046           8.14431E+13          814,430.6137    0.70%     5,701
0.00000047           8.21097E+13          821,096.5740    7.40%    60,761
0.00000048           8.25756E+13          825,755.7494   22.00%   181,666
0.00000049           8.28540E+13          828,539.9525   30.90%   256,019
0.00000050           8.29582E+13          829,581.8860   35.00%   290,354
0.00000051           8.29013E+13          829,013.3247   35.00%   290,155
0.00000052           8.26964E+13          826,963.6505   35.70%   295,226
0.00000053           8.23559E+13          823,558.6947   35.70%   294,010
0.00000054           8.18920E+13          818,919.8475   35.70%   292,354
0.00000055           8.13163E+13          813,163.3952   37.40%   304,123
0.00000056           8.06400E+13          806,400.0523   35.00%   282,240
0.00000057           7.98735E+13          798,734.6540   32.50%   259,589
0.00000058           7.90266E+13          790,265.9847   28.90%   228,387
0.00000059           7.81087E+13          781,086.7161   23.40%   182,774
0.00000060           7.71283E+13          771,283.4340   19.50%   150,400
0.00000061           7.60937E+13          760,936.7362    3.70%    28,155
0.00000062           7.50121E+13          750,121.3857    0.25%     1,875

--
l is wavelength in meters, ((2phc²)/l⁵)/(e^(hc/lkT)-1) is the Planck Radiation Density of each wavelength in watts/m²/m, El_I is the Planck Radiation Density times the wavelength interval (10 nm) in watts, V_J is the Johnson V-band filter for that wavelength, and El_IV_J is the Planck Radiation Density wavelength interval as reduced by the Johnson V-Band filter ratio.

The sum of all Planck Radiation Density·wavelength interval across the entire spectrum is equal to the total energy output of the star. (confirmed against the same data points using P=seT⁴) Thus, the same frequencies through the Johnson V-band filter should show us the total energy as seen through a Yellow Filter. That is a visual flux of 3,403,790 watts/m² summing all the wavelengths above. Solving for 1 AU distant using inverse square we get 73.57 watts/m². Using the same data to derrive Absolute Magnitude by M=4.85-(LOG₁₀( (Flux_star x Area_star) / (Flux_sol x Area_sol) ) x 2.5

Since this is sol, that's Log₁₀ ( (3,403,790w x 6.07874777E+18m²) / (3,403,790w x 6.07874777E+18m²) ) or Log₁₀(x/x) or Log₁₀(1), which is 0.

4.85 = 4.85 - (0 x 2.5)

Plugging in Sirius A we get:
(25,297,303w x 1.77977420E+19m²) / (3,403,790w x 6.07874777E+18m²) = 21.7601694
Log₁₀(21.7601694) x 2.5 = 3.344
4.85 - (3.344) = 1.506 » 1.47

We're within 1/27^th of an order of magnitude, so that's pretty good. But plugging in Sirius B we get:
(190,470,268w x 4.53776090E+14m²) / (3,403,790w x 6.07874777E+18m²) = 0.004177264
Log₁₀(0.004177264) x 2.5 = -5.948
4.85 - (-5.948) = 10.798 ¹ 11.34

Off by over half an order of magnitude. That's a difference of over 50% luminosity.

So... do I just not get filter throughput function, or is there something else I'm not doing?

 

[SpaceTiger]

Off by over half an order of magnitude. That's a difference of over 50% luminosity.

You've gone to a lot of trouble to check your results, so I suspect you've done your calculations right. Several other things to consider:

- There are multiple definitions of the "V" band (even more than one associated with Johnson), so the one you've modeled may not be exactly the same as the one that was used for that table.
- White dwarfs are not perfect blackbodies and actually show broad absorption lines in the optical.
- A very small fraction of the bolometric output of a white dwarf is in the visible range (as you already noted), so deviations from the blackbody prediction would be more severe than for, say, the sun.

I still wouldn't expect the blackbody prediction to be off by that much, but I've never done the exercise myself, so I'm not sure. The fact that you get reasonable results for the sun and Sirius A (neither of which will be perfect blackbodies either) suggests to me that you're doing the right thing.

 

[Chronos]

It should be possible to more or less directly calculate the bolometric and absolute magnitudes based on effective temperature and size of the star. Here is a calculator that does just that:

http://www.go.ednet.ns.ca/~larry/astro/HR_diag.html

Sirius A [R=1.711; T = 9900] yields
M[bol] = 1.247
Mv = 1.668

Sirius B [R = .0084; T = 25193]
M[bol] = 8.735
Mv = 11.401

Sun [R = 1; T = 5800]
M[bol] = 4.735
Mv = 4.910

 

[-Job-]

You guys are pretty hardcore. :P

 

[Fallguy]

You've gone to a lot of trouble to check your results, so I suspect you've done your calculations right. Several other things to consider:

- There are multiple definitions of the "V" band (even more than one associated with Johnson), so the one you've modeled may not be exactly the same as the one that was used for that table.

This is what I expected... that Johnson V-band was not a set standard of frequency filters, but a kind of filter. How do astronomers keep it all straight if the differences can be this huge?

- White dwarfs are not perfect blackbodies and actually show broad absorption lines in the optical.

My math includes non-perfect emission. (esT⁴ where e is the blackbody emission function) I used proper H_abgd and other appropriate absorption spectra, based on stellar class. Within a very close order of magnitude, all three stars in question have e values at or around 0.9994. Not perfect blackbodies, but very close. Sirius B is a class DA2 white dwarf, meaning it has very strong Balmer hydrogen spectral lines. These reduce its blackbody emission slightly in the visual spectrum, but about the same as for its companion, Sirius A. (which, being a Type A0V, also has strong Balmer hydrogen spectral lines)

- A very small fraction of the bolometric output of a white dwarf is in the visible range (as you already noted), so deviations from the blackbody prediction would be more severe than for, say, the sun.

True, but the T_eff and Radius are well documented. Although DA2 white dwarfs do emmit the vast majority of their energy well above the visual spectrum, the curve is still very predictable and should result in more accuracy than I'm getting. I'm beginning to highly suspect that the V-band filters used in the RECONS data skew the results badly for white dwarfs. I have plugged in a few other binary star systems (Alpha Centauri, Procyon, and a few others) and get the same results. Without applying a filter, but just using total energy within the visual spectrum, all non-white dwarf stars are within ±0.01 magnitude of the RECONS data. That's well within the margin of error of the observations, so I'm looking good.

The exception is Procyon B. (wouldn't you know it?) It's off by +0.09 magnitude, so we're within a close order of magnitude but still off. But this time it's the other way, too high, not too low. (Sirius B is off by -0.54)

I still wouldn't expect the blackbody prediction to be off by that much, but I've never done the exercise myself, so I'm not sure. The fact that you get reasonable results for the sun and Sirius A (neither of which will be perfect blackbodies either) suggests to me that you're doing the right thing.

Have you examined my equations? That's the key to this whole thing, and the reason I'm comparing against the RECONS data, is to see if the equations are done right. The whole point and purpose of this is to confirm my understanding of the math involved in the theories. If the math can replicate observation, then I did it right and my understanding is sound. If it can't, then my math is wrong and I'm not getting something. Since Sirius B was wrong, I began to doubt my understanding of the math and needed confirmation as to where I went wrong.

I am now confident that my understanding is sound and that the math is sound. (unless someone can point out where I am misunderstanding something) With that in mind, it leaves the question...

What is it about White Dwarfs in the RECONS database that skews their magnitude so far off of prediction?

It should be possible to more or less directly calculate the bolometric and absolute magnitudes based on effective temperature and size of the star. Here is a calculator that does just that:

I can already calculate bolometric magnitude with a high degree of accuracy using my spreadsheet that I built for that. My problem is the RECONS data not being reproducable with white dwarf stars.

Hmm... I just noticed that at the bottom of the RECONS database regarding white dwarfs they assume a stellar mass of 0.5 for all. Could it be that they're also using a set radius for all white dwarfs in the database? If so, that would explain why Sirius B is so far off... it has a mass of just under 1.0, which translates to a much smaller radius than a white dwarf with mass 0.5. (remembering that white dwarfs decrease in size directly proportional to increases in their mass) Since Procyon B is very close to 0.5 mass, it would thus be closer to the assumed radius and be much closer to the RECONS data. (which it is)

I'll have to look into this further.

You guys are pretty hardcore. :P

Yep. Wanna know the truly sad part? This is just for fun. I don't work in astronomy, physics, or any sort of space-related industry. I'm a computer tech. I'm just doing this to advance my understanding of stellar theory for my own sake. :tongue: