Redshift Galaxy: Calculate Velocity, Distance & Look-Back Time

Understanding the cosmos often involves delving into the fascinating properties of distant galaxies. One crucial aspect is determining their velocity, distance, and look-back time, especially when we know their redshift. Let's explore how to calculate these parameters for a galaxy with a redshift of z = 0.15, using the Doppler formula and Hubble's law.

Understanding Redshift

Before we dive into the calculations, let's briefly discuss what redshift is. Redshift (denoted by z) is a phenomenon where the light emitted by a celestial object appears shifted towards the red end of the spectrum. This happens when the object is moving away from us. The higher the redshift value, the faster the object is receding, and the farther away it is. Redshift is a cornerstone of modern cosmology, providing essential clues about the expansion of the universe.

The concept of redshift stems from the Doppler effect, which is familiar in everyday life. Think of the changing pitch of a siren as an ambulance passes by. When the ambulance approaches, the sound waves are compressed, leading to a higher pitch (blueshift for light). Conversely, as the ambulance moves away, the sound waves are stretched, resulting in a lower pitch (redshift for light). In the context of cosmology, the redshift we observe in distant galaxies is primarily due to the expansion of the universe, which stretches the wavelengths of light as it travels across vast cosmic distances.

In simpler terms, imagine the universe as a balloon being inflated. Galaxies are like dots on the surface of the balloon. As the balloon expands, the distance between the dots increases. Similarly, as the universe expands, the space between galaxies stretches, causing the light emitted by these galaxies to redshift. The amount of redshift is directly related to how much the universe has expanded during the light's journey from the galaxy to us. Therefore, redshift serves as a powerful tool for measuring distances and velocities of distant objects in the cosmos. Understanding redshift is paramount to comprehending the scale, age, and evolution of the universe, making it an indispensable concept in astrophysics and cosmology.

Calculating Velocity Using the Doppler Formula

The Doppler formula relates the redshift (z) to the velocity (v) of the galaxy. For relatively small redshifts (z << 1), which is the case here with z = 0.15, we can use the simplified Doppler formula:

$$v ≈ z × c$$

Where:

• v is the velocity of the galaxy.
• z is the redshift.
• c is the speed of light, approximately $3 × 10^5$ km/s.

Let's plug in the values:

v ≈ 0.15 × (3 × 10^5 "),$ km/s $v ≈ 4.5 × 10^4 ",$ km/s So, the velocity of the galaxy is approximately 45,000 km/s. This means the galaxy is moving away from us at a considerable fraction of the speed of light. It’s important to note that this simplified formula is accurate for relatively small redshifts. For higher redshifts, a relativistic correction is needed to account for the effects of special relativity. In essence, the Doppler formula provides a direct link between the observed redshift of a galaxy and its recessional velocity. This relationship is fundamental to our understanding of the expanding universe and allows astronomers to estimate the speeds at which galaxies are moving away from us. The accuracy of this calculation depends on the redshift value, and for more distant and rapidly receding galaxies, more sophisticated relativistic formulas are employed. ## Determining Distance Using Hubble's Law **Hubble's Law** provides a relationship between the velocity (*v*) of a galaxy and its distance (*d*) from us: $v = H₀ × d

Where:

• v is the velocity of the galaxy (which we calculated above).
• H₀ is the Hubble constant, which represents the rate of expansion of the universe. Its value is approximately 70 km/s/Mpc (kilometers per second per megaparsec). Note: 1 Mpc = 1 million parsecs, and 1 parsec is about 3.26 light-years.
• d is the distance to the galaxy.

We can rearrange Hubble's Law to solve for the distance:

$$d = v / H₀$$

Plugging in the values:

d = (4.5 × 10^4 ",$ km/s) / (70 km/s/Mpc)" $d ≈ 642.86 ",$ Mpc So, the distance to the galaxy is approximately 642.86 Mpc. To convert this to light-years, we multiply by 3.26: $d ≈ 642.86 × 3.26 ",$ light-years $d ≈ 2096 ",$ light-years Thus, the galaxy is approximately 2096 million light-years away. Hubble's Law is a cornerstone of modern cosmology, providing a direct relationship between a galaxy's distance and its recessional velocity. The Hubble constant, *H₀*, is a crucial parameter that defines the rate at which the universe is expanding. However, it's important to note that the value of the Hubble constant is still subject to ongoing research and refinement, with different measurement techniques yielding slightly different results. These discrepancies have led to what is known as the Hubble tension, a significant area of investigation in contemporary cosmology. The distance calculated using Hubble's Law is based on the assumption that the expansion rate of the universe has been constant over time. However, the expansion rate has actually varied throughout the universe's history. Therefore, the distance calculated using Hubble's Law is a