MOND -- Understanding Equations

[Arman777]

I am reading an article written by Sanders and McGaugh In the article, the first equation is written as $$F = f(r/r_0)GM/r^2~~~(1)$$

where $x = r/r_0$
Then it goes like if
$$f(x) =
\begin{cases}
1 & \text{if } x <<1 \\
x & \text{if } x >>1
\end{cases} $$

So the equation becomes

$$F =
\begin{cases}
GM/r^2 & \text{if } x <<1 \\
GM/rr_0 & \text{if } x >>1
\end{cases} $$

Then he defines the force acting on the particle $m$ as

$$F = ma\mu(a/a_0)$$

However in the wikipedia its claimed that$$F = \frac {GMm} {\mu(a/a_0)r^2} $$

My first question is what is this $F$ ? It cannot be force since the units do not match. Is it acceleration ?

Or in (1) $m$ is taken as 1 ?

The wiki equation and the equation (1) are the same ?

 

[kimbyd]

$F$ is likely the gravitational field, which is the force per unit mass of the object the force acts upon.

 

[Arman777]

$F$ is likely the gravitational field, which is the force per unit mass of the object the force acts upon.

I see. Then should we use $\mu(a/a_0)$ or $\mu(r/r_0)$ ? Is there a difference ? For instance if I write $$F = f(r/r_0)GM/r^2=f(a/a_0)GM/r^2$$ Is this true ?

 

[kimbyd]

I see. Then should we use $\mu(a/a_0)$ or $\mu(r/r_0)$ ? Is there a difference ? For instance if I write $$F = f(r/r_0)GM/r^2=f(a/a_0)GM/r^2$$ Is this true ?

I think the difference between the two is just notation. You can always re-express a function in terms of different parameters if you want. With the above, if $F$ is a function of $r$, then the second equation written out fully would be:
$$F(r) = f(a(r)/a_0)GM/r^2$$

 

[Vanadium 50]

In the article, the first equation

Did you actually read the paper? The first equation is not MOND, and the first two pages of the paper explain why it's not MOND.

 

[Arman777]

Did you actually read the paper? The first equation is not MOND, and the first two pages of the paper explain why it's not MOND.

I was reading but I missed that sentence I think. Okay thanks

 

[Vanadium 50]

I missed that sentence I think

first two pages

Two pages are more than a sentence. When you find yourself in a hole, it's best to stop digging. Given that this whole thread is based on reconciling two equations that aren't even supposed to be the same, I am going to ask it be closed.

 

[Doc Al]

On that note, thread closed.