# A NEW APPROACH

## INTRODUCTION

Velocity is Relative

## THE NEW APPROACH

A short review on the concept of inertial frames in classical mechanics

A possible explanation for a well known discrepancy

## RESULTS

Fits to observed data; nice features ## Introduction

Velocity is relative. When one states: "I am moving at a rate of 2 meters per second", one actually assumes that there is some reference point, relative to which he is changing the distance. In exactly the same way, when an astronomer states: "this gas cloud is rotating around the galactic center at a rate of 80 Km/sec", she actually means that she has been supplied with a well defined frame of reference, relative to which this value is valid. The observed values presented in rotation curves are also valid only with respect to some specific frame of reference. The only question is whether this frame is an Inertial one.

Why is it relevant? Because currently, when fitting rotation curves, astrophysicists take this quality for granted. Modeling a rotation curve (from the corresponding mass distribution) implicitly relies on this assumption. We shall come back to this point later, but first let's define what an Inertial frame is.

## Inertial Frames

Strictly speaking, an Inertial frame is a frame of reference relative to which the Law of Inertia holds. The law of Inertia (i.e. the first law of Newton) states that an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force. This law is quite "natural", especially when dealing with daily life activities relative to our "stationary" ground frame. Take for instance an Olympic ice dancer. While ice skating, she expects to keep her constant velocity if no obstacles are present. While going to sleep at night, she expects to stay in her bed...

That's when no forces are in action. What happens though, when an object is acted upon by a force? In this case the trajectory can be found by using Newton's second law.

But not all frames are Inertial. Relative to some frames, free objects follow an accelerated or a curved trajectory even though no forces are present. Observing the physical reality relative to such frames is counter intuitive. Take a look, for instance, at the excellent demonstration below. Not only is this strange but it also contradicts the law of inertia. A free body (not acted upon by forces) does not keep its constant velocity. In order to overcome this issue on the one hand while preserving Newton's laws on the other hand, sixteenth century physicists introduced the concept of fictitious forces. These forces were invented as the cause (i.e. the explanation) for the curved trajectories of free objects. However, the origin of these forces was unclear. It was not until Mach and Einstein that a more general understanding of the subject has emerged. In any case, fictitious forces are a necessary and a useful feature in classical mechanics. They are fictitious only in the sense that we cannot perceive their origin. In all other aspects they are real.

Let us sum up: a natural phenomenon can be described (by equations) relative to any frame of reference. In classical mechanics we are often interested in the trajectories of objects. While relative to Inertial frames the trajectory of a body can be found by using Newton's second law (which is reduced to constant-velocity motion when no forces are in action), the situation relative to non-Inertial frames is a bit different. Relative to non-Inertial frames Newton's second law can be used only when fictitious forces are added to the equation.

## The New Approach Explained

The new approach is actually a new model for the rotational velocities (i.e. a prediction). The core idea behind this prediction is the insight that the observed velocities might be given relative to a non-inertial frame. Relative to such a frame, as we already know, fictitious forces are in action. If those are to be taken into account within the model it might be possible to explain the gap. But before we dive more into details, let us start from the beginning. How do we measure the rotational velocities in galaxies?

The first step in our journey is observing a galaxy. Well... one may try it with his own naked eyes. But then, in order to meet current standards, one must: 1. have two radio-sensitive eyes,  2. be willing to stare at the same point for a very long time. Radio telescopes are just perfect for this mission. Therefore they are widely used for the extraction of rotation curves. ## *** For the direction and magnitude of the fictitious forces (which are actually not necessary for the fitting process) please download our paper. These forces should be equal to the gravitational forces produced by a dark halo. ## Frames of Reference

The measured velocities presented in RC's are valid only with respect to K' by definition. If K' is not Inertial, then the relative motion between K (the inertial frame) and K' (the observational frame) includes only rotation. The relation between the rotational velocities in the two frames is given by: Vk'(r) = Vk(r) + * r

## Results

In the previous section we introduced a possible explanation for the discrepancy. The bottom line can be described as follows: the new model for the rotational velocities V(r) should include an extra term (namely w * r). This extra term arises in non-inertial systems. It represents the additional tangential velocity relative to those systems. The motivation for using such a term comes from the simple insight that the observed velocities might actually be non-inertial.

In the following graphs we introduce the fittings of several different rotation curves. More details on the fitting process and the sample of galaxies can be found in the paper. Note that when plotting a graph such as a rotation curve (i.e. something vs radius), an w * r term is simply represented by straight line.

## NGC 300

In this fit a value of w = 1.55 [10^-16 rad/sec] and a value of M/L = 0.97 [Msun/Lsun] were obtained.

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The blue curve represents the prediction relative to a local Inertial frame (based on the visible matter alone). The green line is the extra term required in non-inertial frames and the red curve is our model's prediction.

## Therefore, we hereby declare on the Century's Competition! We offer a 3.1415 \$ prize to the first person to give a satisfactory answer to one of the questions above. The prize will be collected from the International Institute of Pure Science. On a slightly more serious note, we express our deepest hope that the idea we present here will lead to a broader look of the current problems in astrophysics and inspire future scientists. ©2019 by Galaxy Rotation Curves. Proudly created with Wix.com