How to measure distances to stars by means of spectroscopic parallaxes on practice? What is the accuracy of measuring distances using this method compared with distances based on HIPPARCOS trigonometric parallaxes? My sample of objects consists of A-B type stars.
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2 Answers
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Actually the method described on Wikipedia is not the method that is meant by Spectroscopic Parallax. To determine the spectroscopic parallax, you'll need a spectrum of the star and measure the widths of the spectral lines. Compact stars have higher surface gravity, which means that their spectral lines are broader. This means that the radius of the star can be calculated from line widening. Using the observed effective temperature $T_{eff}$ and the radius R, you can get the Luminosity from Stefan's Law, which gives an absolute magnitude and together with the apparent magnitude, a distance modulus and therefore a distance. This gives an accuracy of 10% in luminosity and 5% in distance for main sequence stars and 25-30% in distance for giant stars. (see de Grijs, 2011)
Wilson and Bappu (1957) proposed to use the width (W_0) of the emission core of the CaIII absorption line at 3933 Angstrom. Using data from Hipparcos the relation between the absolute magnitude in V and the width of the absorption line was calibrated to:
$M_V = 33.2-18.0 \log(W_0)$
by Pace er al. (2003). So to use this method you would need to be able to measure the width of the CaIII absorption line, but other calibrations using the MgIIk line also exist.
As these distance measurements are calibrated using Hipparcos, they will never be able to be more accurate until better calibration data is available (i.e. with Gaia).
The much more simpler method as described on Wikipedia might also be used but that is even less accurate. NB. The relation between the colour (colour index B-V) and the spectral class is not linear. Many Hertzsprung-Russel diagrams show both colour and spectral type on the X axis but this is not really correct. A K0Ia supergiant has a different colour index than a K0V star on the main sequence. On page 50 in de Grijs, 2011, this relation is shown in Fig 2.11. It should also be somewhere on Wikimedia (to which it is attributed) but I cannot find it. EDIT: Here it is: http://en.wikipedia.org/wiki/File:H-R_diagram.svg
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If you really want to use the method outlined in Wikipedia than you might use the Hipparcos catalogue to determine the mean absolute magnitude for each spectral type.
I tried this and I got the following values for A and B type main sequence stars:
Spec N M_V <d_err> max(d_err)
B0V [23] -1.8 ± 2.3 130 % 1014 %
B1V [95] -2.1 ± 1.8 81 % 833 %
B2V [223] -1.6 ± 1.5 69 % 1562 %
B3V [279] -1.3 ± 1.4 59 % 739 %
B4V [91] -1.1 ± 1.1 44 % 232 %
B5V [251] -0.8 ± 1.2 44 % 465 %
B6V [145] -0.6 ± 1.3 53 % 881 %
B7V [122] -0.5 ± 1.4 70 % 1405 %
B8V [522] -0.2 ± 1.3 58 % 1988 %
B9V [887] 0.4 ± 1.2 57 % 4613 %
A0V [1206] 0.8 ± 1.2 54 % 2990 %
A1V [735] 0.9 ± 1.1 50 % 2509 %
A2V [491] 1.0 ± 1.0 42 % 1076 %
A3V [451] 1.1 ± 1.1 51 % 3526 %
A4V [178] 1.3 ± 1.1 49 % 1106 %
A5V [214] 1.3 ± 1.1 45 % 762 %
A6V [99] 1.4 ± 1.1 54 % 1924 %
A7V [188] 1.6 ± 1.2 65 % 5371 %
A8V [114] 1.7 ± 1.0 42 % 392 %
A9V [329] 2.0 ± 1.2 57 % 2788 %
where Spec is the spectral type, N is the number of stars in the Hipparcos catalogue with that spectral type (nb. many stars do not have a spectral type, or a spectral type with the star type (e.g. main sequence or supergiant) missing), the average and standard deviation of the absolute magnitude (M_V), the average error in distance in percentages() and the maximum error in distance (max(d_err)).
NB. I did not take extinction into account!
So you can use these values to calculate the distance to a main sequence A or B type star.
$D = 10^{(m-M_V)/5+1}$
But as you can see the errors are quite large. For A/B type supergiants, even the average error is more than 100%. The lowest average errors I found (25%) are for F type sub giant stars (type IV).