Determining Distances in Space

There are several techniques for measuring distances in space:

1. Geometric Methods  –For close stars

2. Spectroscopic Measurements –Close to medium distance stars (most of the stars in our galaxy)

3. Use of “Standard Candles” –For very bright objects in distant galaxies

4. Using Redshifts. 

Geometric Methods

1)Parallax: - It provides the distance in parsec pc (parallax arcsecond).

2)Angular Size

 If we know the size of an object and what angle it takes up in the sky, then we can calculate how far it is.

 
When D>>d , 𝜹rad ≈ (d/D)
                    𝜹degree ≈ (180°/π)(d/D)
                    𝜹degree ≈ 57.2958°(d/D)
                    𝜹'arc min ≈ 3438' (d/D)
                    𝜹''arc sec ≈ 206265 (d/D)

                     In case of moon, 
                                                          𝜹 = 0.5° or 31.08 arc min or 1865.18 arc sec
                            Radius of moon (d/2) = 1732 km
                                                           d  =  3476 km
          Therefore, Distance of moon (D) = (3476 ✕ 206265) /1865.18
                                                               =384400 km

     Spectroscopic Measurements

If we know an object’s apparent magnitude, and its absolute magnitude, we can calculate how far away it is.

Stellar “Candles”

For distances which are too large, astronomers use 'standard candles'. Light sources which are further away appear fainter because the light is spread out over a greater area.  If we know how luminous a source really is, then we can estimate its distance from how bright it appears from Earth. The light which reaches Earth has spread out over a sphere.

Radius of sphere = distance to earth, r

Surface area of sphere = 4πr ²

On Earth, the received power per unit area is then

received power per unit area = source luminosity/area of sphere 

                                           P = L /4πr ²

Using Redshifts

An object that is redshifted will have its peak brightness appear  towards the red end of the spectrum.

This is calculated with an equation, 

z = (λ_observed - λ_rest)/λ_rest  where

z is redshift parameter.

λ_observed is the observed wavelength of a spectral line.

λ_rest is the wavelength that line would have if its source was not in motion.

z tells the number of years the light from the object has traveled to reach us, however this is not the distance as the universe has been expanding as the light traveled and the object is now much farther away.

z	Time the light has been travelling	Distance to the object now
0.0000715	1 million years	1 million light years
0.10	1.286 billion years	1.349 billion light years
0.25	2.916 billion years	3.260 billion light years
.5	5.019 billion years	5.936 billion light years
1	7.731 billion years	10.147 billion light years