Calculating Galaxy Distances with High Redshift: A Detailed Guide

Introduction

Calculating the distance to a galaxy with high redshift has long been a challenge for cosmologists and astronomers. Understanding the implications of redshift and how to use it to determine the distance to galaxies is crucial in our quest to understand the vast scale of the universe. This article delves into the complexities of using redshift to calculate distances and the role of Hubble's Law in this process.

Understanding Redshift and Its Limitations

Redshift in galaxies is often misunderstood as a result of the Doppler effect, which is commonly used to measure velocities in everyday situations. However, in the context of cosmology and the vastness of space, redshift is often not a direct measure of velocity due to the expansion of space itself. Due to this, redshift primarily indicates the expansion of the universe, not the motion of galaxies relative to us.

The Hubble Law, which suggests a linear relationship between the recession velocity of galaxies and their distance, has been a cornerstone of cosmology. However, the actual value of the Hubble constant (H) has shown some discrepancies, with values ranging from 67 to 73 km/sec per megaparsec. Until this discrepancy is resolved, the accuracy of calculated distances based on redshift remains uncertain. Therefore, for galaxies with high redshifts, the distance calculation can only achieve an accuracy of /- 15%.

Using Redshift to Calculate Distances

To calculate the distance to a galaxy with a known redshift, one can utilize Hubble's Law, which states that the recession velocity (v) of a galaxy is directly proportional to its distance (d) from us:

Hubble's Law: v cz Hd
where v is the recession velocity in km/sec, c is the speed of light, z is the redshift value, and H is the Hubble parameter in km/sec per megaparsec.

It's important to note that the redshift value needs to be corrected for the Earth's orbital velocity around the Sun (to obtain a heliocentric measure) and the Sun's orbital velocity around the galactic center (to obtain a galactocentric measure).

Example Calculation

Let's consider a galaxy with a redshift of 7. Using Hubble's Law:

v cz Hd
v 7 * 300,000 km/sec (since c 300,000 km/sec)
v 2,100,000 km/sec

Dividing the recession velocity by the Hubble constant (70 km/sec per megaparsec):

d v / H 2,100,000 km/sec / 70 km/sec per megaparsec
d ≈ 300 megaparsecs

Converting megaparsecs to light-years (1 parsec ≈ 3.26 light-years):

300 megaparsecs ≈ 300 * 3.26 * 1,000,000 light-years ≈ 978,000,000 light-years

Alternative Measures in Cosmology

While redshift is a primary method for calculating distances, there are other measures in cosmology that are used, such as angular distance and luminosity distance. These measures are similar for small redshifts but have different formulae for large redshifts.

Angular Distance: This is the angular separation between two celestial objects as seen from Earth. It is often used to compare the apparent positions of galaxies.

Luminosity Distance: This is a more accurate distance measure that takes into account the dimming of light due to the expansion of space. It is particularly useful for extremely distant galaxies where other measures may be less reliable.

In conclusion, while redshift is a powerful tool for understanding the vast distances in the universe, its use for distance calculations, especially for high-redshift galaxies, is subject to limitations due to the current uncertainties in the Hubble constant. Utilizing Hubble's Law and understanding the corrections needed for various measures can enhance the accuracy but do not eliminate the inherent uncertainties in these calculations.