# How to calculate the strength ("space force") of dark matter within galaxies?

Let's first have a look at the observed speed within galaxies (source):

On the left side we can see the speed of celestial bodies within galaxies derived by gravity only (gravity is usually created by a black hole at the center of a galaxy), on the right side we can see the observed speed of celestial bodies, both in respect of the distance from the galactic center.

As a result, the strength of dark matter (or "space force") could be calculated as:

$$F$$dark matter = $$m$$galaxy $$\cdot$$ $$v$$celestial body within galaxy $$\frac{d}{dt}$$ - $$g$$black hole, center of galaxy

Is this a viable calculation? And if so, are there any a graphs online which show this strength of dark matter (or "space force") in respect of the distance from the galactic center?

(Note: In this formula, a galaxy would be assigned a mass $$m$$galaxy much higher than the mass of its black hole. However, mathematically speaking one could also just decrease $$dt$$, so with a celestial body wandering from the center to the outskirts of the galaxy, $$dt$$ would decrease, hence the speed of time would increase. I doubt this is true because it would change the basic definition of speed $$v$$celestial body within galaxy as well and introduce all kinds of illogical contradictions, but in math everything is possible as we know from theory of relativity...)

• $$m$$galaxy $$\cdot$$ $$v$$celestial body within galaxy $$\frac{d}{dt}$$ $$[kg·\frac{m}{s^2}]$$ = observable force (shown in picture on the right)
• $$g$$black hole, center of galaxy $$[kg·\frac{m}{s^2}]$$, the force caused by gravity of a black hole (shown in picture on the left)
• $$F$$dark matter $$[kg·\frac{m}{s^2}]$$, the force of dark matter up until a point in space
• Please explain your notation. What do $F_{\text{dark_matter}}$, $dt$ and $G_{\text{black hole}}$ mean ? Without knowing what your terms mean, it is not even clear that your formula is dimensionally correct. Oct 6, 2020 at 13:39
• $mvd/dt$ is the force that is supposed to be caused by dark matter up until a point within the galaxy (shown in the right picture). $G_{black−hole}$ is the force that is caused by the black hole's gravity (shown in the left picture). $F_{dark-matter}$ is the difference... Oct 6, 2020 at 14:19
• Hope that addendum will clarify things... Oct 6, 2020 at 14:29
• "Note that the gravity usually comes from a black hole at the center of a galaxy)". No it doesn't. Oct 6, 2020 at 19:27
• @Rob: Changed some notations, I think it's pretty clear now. Oct 7, 2020 at 7:50

This is pretty much what the velocity/distance curves show. The difference between observed and theoretical gravitational force can be worked out from the rotational velocity at a given distance because the centripetal force required to keep a star in a galactic orbit with speed $$v$$ and radius $$r$$ is proportional to $$\frac {v^2}{r}$$.
The super-massive black hole which lies, we believe, at the centre of most galaxies makes only a very small contribution to the gravitational attraction of the galaxy as whole. The typical mass of such a black hole is around $$1 \%$$ of mass of the stars in the galaxy, or around $$0.1 \%$$ of the total galactic mass including dark matter.