B2 - spectroscopic parallaxes or main-sequence fitting

it does not actually involve parallaxes
based on the following hypotheses:
- a calibrated HR diagram
- can measure spectral colour of states on the HR diagram
- the colours do not change with distance, but $F \propto d^{- 2}$
- spectrum of a known source (star A) with known position on the main sequence (MS)

image: H Cegla, lecture notes

the unknown source, star B, has the same spectra as star A, just different fluxes
for star B, $T_{e f f /}$ or colour is known, but not $L$ and $d$
by measuring the colour/spectrum and reading the position on the HR diagram, its $L$ , and therefore its $M$ can be known
using the distance modulus, its $d$ can be known
this can be extended for stellar clusters too
cluster 1 maps out the MS, $d_{1}$ and $M_{1}$ are known, and assume the inter-cluster distance is small
cluster 2 also maps out the MS, but $d_{2}$ and $M_{2}$ are unknown, $m_{2}$ is known
since it is also on the MS, $M_{1} = M_{2}$
using the distance modulus, $d_{2}$ can also be found
the key hypothesis is that all MS have a given $L$ from the HR diagram and are essentially the same type of stars
using step 2 of the distance ladder, the first estimate of the Milky Way was made in early 1900's to be $7000 \times 1300$ pc (greatly underestimated!)
the main reason is the violation of the central hypothesis that colours do not change with distance
colours do change with distance due to differential absorption of radiation by dust, or differential extinction