The Universe Is Expanding — Question 4

Question 4

The slope of a Hubble diagram can be estimated without fitting an intercept

Assume negligible distance errors, equal velocity errors and the model $v_i=HD_i$ through the origin. Three galaxies have distances 10,20,30 Mpc and velocities 720,1380,2130 km/s.

1. Derive the least-squares slope and calculate H.

2. Find all residuals and their squared sum.

Question 4 - Solution

The slope of a Hubble diagram can be estimated without fitting an intercept

(a) Method and calculation

$$\frac{d}{dH}\sum(v_i-HD_i)^2=-2\sum D_i(v_i-HD_i)=0.$$$$\boxed{H=\frac{\sum D_iv_i}{\sum D_i^2}=\frac{98700}{1400}=70.5\ {\mathrm{km\,s^{-1}\,Mpc^{-1}}}}.$$

(b) Interpretation and further calculation

$$\boxed{r_i=(15,-30,15)\ {\mathrm{km/s}}}.$$$$\boxed{\sum r_i^2=225+900+225=1350\ ({\mathrm{km/s}})^2}.$$

Physical check

An intercept, distance uncertainty, selection effects or correlated peculiar motions require a more complete fit.