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To measure the velocity of a star (away from us), we use its Doppler redshift. However, a redshift may be a result of other factors, such as different gravitational potential (which is always the case, as the light is emitted from the surface of a star and received here on Earth). So how do we single out the redshift which is due to the Doppler effect?

It seems that we must have a way to measure the star's mass and radius, before we can measure its (relative) speed. How is it done? I think the radius cannot be measured directly from Earth, if the star is far enough (that is, if all we see is a dot).

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Indeed, the relative radial velocities quoted for stars do not usually separate out that component which is contributed by a General Relativistic shift caused by time dilation in the potential of the star.

The size of this GR shift, in terms of an equivalent Doppler velocity, is given to a good approximation by $GM/Rc$, where $M$ and $R$ are the stellar mass and radius.

Putting this in "solar units", the GR velocity shift is $$ V_{\rm GR} \simeq 0.6 \left(\frac{M}{M_\odot}\right)\left(\frac{R}{R_\odot}\right)^{-1}\ {\rm km/s}\ .$$

In principle, one could correct radial velocities for this shift, but in practice, the masses and radii of most stars are not known. In a fortunate coincidence, the masses and radii of main sequence stars are correlated ($R \propto M^\alpha$, with $\alpha \simeq 1$) so that the size of the shift does not vary much across the main sequence. However giant stars should have a much lower shift ($\leq 0.1$ km/s). "Errors" at the level of 100s of m/s do not make much difference to most work on stellar dynamics because (a) that is at or below the limits of precision in most spectroscopy and (b) the velocity dispersions of stars in the Galaxy is measured in the range 10-100 km/s.

Measurement of the masses and radii of stars is a broad and complex topic. Radii can be measured in eclipsing binary systems or by use of interferometers for stars with large angular radii. Mass estimates also come from binary systems. These measurements can then be used to calibrate secondary relationships with luminosities and temperatures.

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  • $\begingroup$ So that commonly V listed = V gr + V std doppler? $\endgroup$
    – Alchimista
    Dec 14, 2021 at 10:16
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    $\begingroup$ @Alchimista yes. Almost always. $\endgroup$
    – ProfRob
    Dec 14, 2021 at 13:38

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