# How to measure distances to stars by means of spectroscopic parallaxes?

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?

• This would be better suited for astronomy.stackexchange. – Sandesh Kalantre Jan 26 '14 at 2:17
• This question appears to be off-topic because it belongs on astronomy.stackexchange.com – John Rennie Jan 26 '14 at 9:26
• Just to be clear, astronomy and astrophysics are decidedly on-topic for this site. Questions about amateur observing techniques might get a better audience on the other site, and this site doesn't do recommendations for what commercial products to purchase, but everything else is fine here. – user10851 Feb 1 '14 at 1:45

Spectroscopic parallax is the technique whereby you estimate the absolute magnitude (i.e. the brightness it would have if it were placed at 10 pc) by estimating what "type" of star it is using information fro a spectrum.

It can be applied to any kind of star where (a) you have a reasonable chance of determining the type of star from its spectrum and (b) where there is a single valued relationship between absolute magnitude and spectral type that has been calibrated using stars at known distance (for example stars in clusters or stars with accurate trigonometric parallax).

Once an absolute magnitude is established then there is a straightforward relationship between apparent magnitude and absolute magnitude $$m - M = 5 \log d - 5,$$where $m$ is the apparent magnitude, $M$ is the estimated absolute magnitude and $d$ is the distance in parsecs.

So, where do the uncertainties arise:

(1) There could be extinction along the line of sight. This would have to be included in the equation above. If one has a flux-calibrated spectrum one could try to estimate the extinction from that or fro a comparison of the stars measured colours with the unreddened expected colours for a star of that spectral type.

(2) Spectral type estimation can be uncertain. Even in the best cases it is usually uncertain by $\pm 1$ MK spectral subclasses. How that translates into an uncertainty in absolute magnitude depends on what kind of star you are looking at.

(3) Calibration uncertainty. The relationship between spectral type and absolute magnitude must be calibrated, but it likely depends on the chemical composition of the star. It could be that you can estimate this from your spectrum, but there will still be a calibration uncertainty.

(4) Binarity. It is vital that the star you are looking at is not a binary! Unresolved binarity could cause a mis-estimate of the spectral type and could also mean that the star is apparently brighter than a single str of that type should be. For example, two equal stars which are unresolved will have a spectrum that yields the correct spectral type, but the object will be twice as bright, leading to a $\sqrt{2}$ error in the distance estimate.

It is difficult to generalise, especially as you have not specified how accurate the trigonometric parallaxes are that were are comparing with. However, I will.

Parallaxes for nearby stars can be as good as 1% - that is the distances are known to 1%, and this is due to be improved shortly with data from the Gaia spacecraft, for many millions of the nearest stars. In the best cases - good spectrum, main sequence star, little extinction, no evidence for binarity, you are probably talking about estimating the absolute magnitude to about $\pm 0.2$ mag for solar type stars and hotter, perhaps increasing to $\pm 0.5$ mag for the coolest stars (see this table for an example calibration). Hence this is equivalent to an error in $\log d$ of 0.04 to 0.1 dex and hence an error in distance of $\pm$10-25%

Any influence of observational uncertainties, or spectral types beyond the main sequence will increase this uncertainty.