Tully–Fisher relation

Big space clouds are called galaxies.

Some galaxies spin very fast. Others spin more slowly. Fast galaxies are also very bright.

We can use this to learn a lot. We see how bright a galaxy looks to us. Then we look at how fast it spins. This tells us how far away it is. It is like a tool for space. It helps us map the deep sky. It is a way to find distant things.

Spiral galaxies are huge groups of stars. They spin like giant plates. Scientists found a special way to measure how far away they are. This is called the Tully–Fisher relation.

This rule connects two main things. First, it looks at how fast a galaxy spins. We can measure this by looking at hydrogen gas. Second, it looks at luminosity. Luminosity is the total light a galaxy gives off.

In general, bright galaxies spin very fast. Dim galaxies spin more slowly. We can see how bright a galaxy looks from Earth. But that brightness changes with distance. By measuring the spin, we can find the true luminosity. Then we can figure out the distance. It is like a step on a cosmic ladder.

Some scientists study baryonic mass. This is the total mass of stars and gas. They call this the baryonic Tully–Fisher relation. This version of the rule is very strong. It helps us understand how dark matter works too. Dark matter is a hidden mass that helps hold galaxies together.

Astronomers use a special rule to find how far away galaxies are. This rule is called the Tully–Fisher relation. It helps us measure the distance to spiral galaxies. This is important because galaxies are very far away. We cannot use a ruler to measure the space between stars. Instead, we look for clues in how galaxies behave. The Tully–Fisher relation acts like a step on a cosmic distance ladder.

This rule works by connecting two different things. First, it looks at how fast a galaxy rotates. We can measure this by looking at the 21cm hydrogen line. This line tells us the velocity dispersion of the spiral arms. Second, it looks at the intrinsic luminosity, which is the true brightness of the galaxy. Usually, the brighter a galaxy is, the faster it spins. We can see how bright a galaxy looks from Earth. But that brightness changes depending on the distance.

People have studied these connections for a long time. In 1922, Ernst Öpik used rotation and distance to study the Andromeda Galaxy. This helped prove that Andromeda was its own separate galaxy. Later, in the 1970s, Balkowski and others studied 13 galaxies. They used the data to look at different galaxy shapes. Finally, R. Brent Tully and J. Richard Fisher published their specific relation in 1977. They used groups like the Local Group and the M101 Group to calibrate it.

There are many ways to use this math today. One big project is called Cosmicflow. The Cosmicflow-4 catalog includes data for 10,000 galaxies. Scientists have used this to find many values for the Hubble constant. There are also different versions of the rule. The stellar mass Tully–Fisher relation uses a galaxy's total stellar mass. The strongest version is the baryonic Tully–Fisher relation, or BTFR. This version uses baryonic mass, which is the sum of stars and gas.

This relation helps us understand the invisible parts of space. Many scientists believe a dark matter halo holds galaxies together. The mass of this dark matter might determine how fast a galaxy rotates. This makes the Tully–Fisher relation a way to see the link between visible and dark matter. Some theories, like Modified Newtonian dynamics, also use this relation. It shows how gravity works at low acceleration. Studying these spinning disks helps us map the whole universe.

The Tully–Fisher relation (TFR) is a vital tool in extragalactic astronomy. It describes a specific link between a spiral galaxy's mass or intrinsic luminosity and its rotation velocity. Intrinsic luminosity refers to the actual, true brightness of a galaxy. Rotation velocity describes how fast the galaxy's disk spins. This relationship is essential because it helps astronomers calculate distances to galaxies. By knowing how fast a galaxy rotates, scientists can estimate how bright it truly is. They then compare that true brightness to how bright the galaxy appears from Earth. This process allows them to determine the distance to that galaxy.

To understand the mechanism, we must look at how light and motion interact. Astronomers measure the width of the 21cm hydrogen line using long-slit spectroscopy. This line width represents the velocity dispersion of the galaxy's spiral arms. Essentially, the width of this signal tells us how fast the galaxy is rotating. Theory suggests that a galaxy's total mass dictates its rotational velocity. This mass also determines the total intrinsic luminosity. Therefore, a more luminous galaxy will generally have a faster rotation. This connection allows the TFR to function as a reliable cosmic distance ladder. It uses direct distance measurements to calibrate itself for use in much larger distances.

There are several distinct versions of this relation based on what scientists measure. The original version used by Tully and Fisher focused on optical luminosity. Later research showed that using microwave to infrared radiation, specifically the K band, provided a tighter relation. This radiation serves as a proxy for stellar mass. Another version is the stellar mass Tully–Fisher relation (STFR). This version replaces luminosity with the total stellar mass of the galaxy. The most accurate version is the baryonic Tully–Fisher relation (BTFR). This version considers the total baryonic mass, which is the sum of all stars and gas. In the BTFR, baryonic mass is proportional to velocity raised to a power of roughly 3.5 to 4.5.

The history of this discovery involves several key astronomers and stages of study. In 1922, Ernst Öpik first used the connection between spectroscopic rotation and distance. He applied this to the Andromeda Galaxy to prove it was a separate galaxy. In the 1970s, researchers like C. Balkowski studied 13 galaxies. They used data to distinguish between different galaxy shapes rather than distances. The formal Tully–Fisher relation was published in 1977 by R. Brent Tully and J. Richard Fisher. They calibrated their correlation using the Local Group, the M81 Group, and the M101 Group. They later used this to find distances to the Virgo Cluster and the Ursa Major Cluster.

The significance of the TFR is seen in modern large-scale surveys. One major collaborative effort is the Cosmicflow project. This project uses Tully–Fisher analysis to create catalogs of galaxy peculiar velocity values. The Cosmicflow-4 catalog has grown to include data for 10,000 galaxies. Because of this data, many different values for the Hubble constant have been derived. These calculations have continued through 2024. The ability to map these velocities helps scientists understand the expansion and movement of the universe.

The TFR also provides deep insights into the nature of matter in space. In the dark matter paradigm, a galaxy lives inside a dark matter halo. The mass of this invisible halo is what primarily determines the rotation velocity. This makes the TFR a way to observe the connection between visible matter and dark matter. Other theories, such as Modified Newtonian dynamics (MOND), offer different explanations. In MOND, the BTFR is a direct result of how gravitational force works at low acceleration. This relation is a key piece of evidence for testing different models of gravity.

Finally, the TFR is part of a larger system of astronomical tools. It is specifically designed for rotationally-supported galaxies like spirals. Other types of galaxies, such as ellipticals, require different methods. For those galaxies, astronomers use the Faber–Jackson relation or the fundamental plane. All these methods work together to help us build the cosmic distance ladder. They allow us to move from nearby stars to the furthest reaches of the observable universe.