# THE DISCREPANCY IN GALAXY ROTATION CURVES

## INTRODUCTION

A short review: on the discrepancy in galaxy rotation curves

## THE ROTATION CURVE OF THE SOLAR SYSTEM

The simple rotation curve of systems in which most of the mass is concentrated at the center.

## THE ROTATION CURVES OF GALAXIES

The mass distributions of spiral galaxies and the possible shapes of their rotation curves.

## THE DISCREPANCY

The tension between expectations and actual results.

## CURRENT SOLUTIONS

The main theories we are using today in order to explain the gap.

## Introduction

The 50-years discrepancy in Galaxy Rotation Curves is in actuality a mismatch between the observed rotational velocities in galaxies and our predictions regarding these velocities. It is customary to plot those velocities as a function of the distance from the galactic center, as can be seen in the graph.

## In order to develop some intuition, let's take a look at this nice visualization. The motions of many point masses are simulated under the influence of their own gravity. At each radius (i.e. distance) the circular velocity takes some value. Plotting these values together forms a rotation curve. (Galaxy Rotation Simulation, Credit: R.Z)

## The very first indications of a strange behavior in galaxy rotation curves dates all the way back to the 1930's (e.g. Babcock 1939). In his paper, Babcock introduces the measured rotation curve of Andromeda (M-31). He is quite surprised by the results, as he states that he had found "..unexpectedly large circular velocities.." (referring to the outer parts of the galaxy). However, this was only one example. It was not until the 1970's that enough data was gathered and the discrepancy became more apparent and a real problem in astrophysics. Vera Rubin (Rubin, Ford & Thonnard 1980), states that "most galaxies exhibit rising rotational velocities... for the very large galaxies are the rotation curves flat". By the end of the 1970's it was clear that the observed rotation curves show an increasing or a flat behavior at the outer parts of a galaxy while Newtonian calculations predict a decreasing curve. The discrepancy could no longer be overlooked.

## In the next two sections we will dive into the theoretical considerations which lead to decreasing Newtonian curves. In the last sections the discrepancy will be discussed further.

## The Rotation Curve Of The Solar System

As a first simple example, let us calculate the rotation curve of our solar system. The following three facts will be helpful: 1. The vast majority of the solar system's mass resides in the sun. 2. Most of the planets' orbits are nearly circular. 3. The gravitational force produced by a spherical symmetrical object (like the sun) on outer bodies is equal to the force produced by a point mass (of the same mass) located at the center of that object.

## This type of behavior, where the velocity decreases with the square root of the radius, is known as a "Keplerian Decline". Note that the velocities in the rotation curve are the actual velocities (usually in km/sec) and not the angular velocities.

## The Rotation Curves Of Galaxies

Unlike the solar system, modeling spiral galaxies is different. As can be seen in the left panel below, the matter within a spiral galaxy is not concentrated only in the center, but is spread out. Moreover, spiral galaxies are flat. The convenient analytical formulas for the gravitational fields within spherical symmetrical objects cannot be used here. It seems that more sophisticated models should be adopted. An example of such a model is the Freeman exponential disk. This is an analytical model for the gravitational potential exerted by a flat disk. The idea here, is to create a rough imitation of real galaxies. The mass-density of the disk drops exponentially as can be seen in the formula below. The gravitational field (as well as the rotational velocities) can then be calculated analytically.

## FREEMAN DISK

A Freeman disk is a flat disk with a mass-density given by the top equation. The model includes two parameters: 1. the density in the center of the disk and 2. the characteristic radius (R) where the density drops by 1/e. In his 1970 paper, Freeman derived the analytical formula for the circular velocities of objects rotating on the plane of the disk (see the bottom equation). The I and K are the modified Bessel functions.

## FREEMAN'S DISK CURVE

A characteristic Freeman's disk rotation curve is presented. Note that at large distances the curve starts to show a Keplerian behavior.

To develop some intuition on how different model parameters affect the shapes of rotation curves please visit the fitting tools section and download the Mass Models Interface.

## A Freeman disk is a convenient approximation for the mass distributions of galaxies. Modern methods, in any case, use the observed light distribution in order to derive (numerically) the mass distribution. Let us clarify: setting a relevant mass model is a necessary and a preliminary step when calculating a rotation curve. Only then, by applying Newton's laws, the expressions for the rotational velocities can be extracted. An example for such a prediction is given below.

## The predicted rotation curve of NGC 4100

The red line represents the predicted rotation curve of NGC 4100. This curve is composed of two contributions: the gas contribution (green line) and the stars contribution (blue line). Both components are based on the corresponding observations of their light distributions. Specifically, for the stars component, a M/L value of 0.8 is used.

Credit: the data was taken from the SPARC database.

## The Discrepancy

In the previous section we dealt with galactic mass distributions and the expected rotation curves based on these distributions. We already have a general understanding of this process:

1. Model the mass distribution of a galaxy analytically or numerically.

2. Apply Newton's laws on this model (i.e. use Newton's second law and Newton's gravitational law to find the gravitational field produced by this mass distribution).

3. Calculate the rotational velocities.

The exact details of this process, in any case, are not so important to our discussion. The more important outcome is the declining shapes of rotation curves. As we have already seen, the density of the matter drops significantly at large distances. At some point ("far" enough, where most of the mass is concentrated "deep" in the center), a Keplerian behavior is expected as an approximation. That is, at some point we expect to see declining curves. However, this was not observed. Measured rotation curves are not in fact declining, despite extending measurements to very large distances. This problem is known as the discrepancy in galaxy rotation curves.

## An astronomer scratching his head (1979)

## Current Solutions

At first sight, it seems that there are only two possible solutions for the discrepancy. Either stage 1 is wrong and we missed something important with our mass modeling, or stage 2 is wrong and the application of the fundamental laws of physics is not valid in these cases.

The vast majority of physicists support the first explanation. According to their view, there must be some additional matter in galaxies. This matter cannot be seen (because the visible mass we see is not enough to explain rotation curves) and must be distributed differently from the visible matter (otherwise rotation curves would still be declining). This kind of matter is known as dark matter. Today, dark matter has grown to be necessary in other fields of astrophysics, but back then in the 70's, rotation curves were the only reason for inventing this concept*.

Some physicists, however, support the second explanation. According to their view, the very application of Newton's laws is wrong. These laws were never put to test in galactic scales before. Perhaps the laws are not general enough. Specifically, according to this theory, Newton's second law should take a different form. The proposed generalization is of such a nature that Newton's original law is obtained as an approximation when "normal" accelerations are used. In galaxies, however, the accelerations are so low that the new prediction deviates significantly from the Newtonian prediction (see the MOND original paper).

Both theories have advantages and disadvantages. The main disadvantage of dark matter is the simple fact that it hasn't been discovered (although scientists have been searching for it for decades). As long as this substance is not found, it must be taken as a weakness of the theory. No matter how deep dark matter is integrated in other fields of physics (i.e. cosmology), it is still a speculation. It is an extra assumption we make on the physical reality. This issue, together with other fundamental problems** have led some astrophysicists to seriously doubt the current paradigm. However, the resources allocated to investigate alternatives are still very limited.

The main disadvantage of the MOND theory is its inconsistency with GR (General Relativity). After all, Newton's laws are an approximation of GR, and if you're going to change Newton's laws you need to change GR in some way or another. This is a whole new story. First, GR is the most successful theory we have to date. Second, it is constructed on basic principles. Changing it in order to meet galactic observations without changing the whole shield is a very difficult goal. These heroic attempts have been made by some physicists in the past, but their overall success is unclear.

In this site we wish to offer a third explanation for the discrepancy. An explanation which does not invoke extra matter or the need of changing Einstein's general relativity. For a comprehensive review of the new approach please visit this page. Our current work is focused only on galaxy rotation curves. However, it's not unlikely that the same core idea will be used for the investigation of other discrepancies (e.g. ellipticals, gravitational lensing). We express our deepest hopes that this work will contribute to our understanding of nature.

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* The history of the hidden mass problem is actually more complicated than that. The "desire" for extra mass in cosmology has already existed in the 1970's. An excellent overview can be found here.

** A comprehensive review of the dark matter paradigm and the unproportional credit it receives nowadays can be found in in our website.