Flat Velocity Curve: Newton vs MOND Theory

[zachzach]

Consider a star of mass m at a distance r form the center of a circular disk galaxy. Newton's law: F = GM(r)m/(r^2) where M(r) is the amount of mass inside the radius r. If we consider a uniform galaxy then density (p) is p = M/L where L is the length = 2*pi*r. So M(r) = p*2*pi*r. Setting the force of gravity equal to centripetal force (mv^2/r) you get G*2*pi*p = v^2 or
v = [2G*pi*p]^(1/2) which is a constant. Why do you need MOND theory. To me it seems Newtonian mechanics predicts a flat velocity curve.

 

[Jonathan Scott]

Consider a star of mass m at a distance r form the center of a circular disk galaxy. Newton's law: F = GM(r)m/(r^2) where M(r) is the amount of mass inside the radius r. If we consider a uniform galaxy then density (p) is p = M/L where L is the length = 2*pi*r. So M(r) = p*2*pi*r. Setting the force of gravity equal to centripetal force (mv^2/r) you get G*2*pi*p = v^2 or
v = [2G*pi*p]^(1/2) which is a constant. Why do you need MOND theory. To me it seems Newtonian mechanics predicts a flat velocity curve.

Firstly, your simplified form of Newton's law only applies in certain cases such as when the mass is spherically symmetrical, or like a segment of a sphere along a diameter towards the relevant direction.

Secondly, I don't get your maths for the mass. If the galaxy is of uniform density per area of the disk, the mass inside a given radius would be proportional to the square of the radius. For the mass to be proportional to the radius, the area density would have to vary as 1/r.

 

[FrankPlanck]

Consider a star of mass m at a distance r form the center of a circular disk galaxy. Newton's law: F = GM(r)m/(r^2) where M(r) is the amount of mass inside the radius r. If we consider a uniform galaxy then density (p) is p = M/L where L is the length = 2*pi*r. So M(r) = p*2*pi*r. Setting the force of gravity equal to centripetal force (mv^2/r) you get G*2*pi*p = v^2 or
v = [2G*pi*p]^(1/2) which is a constant. Why do you need MOND theory. To me it seems Newtonian mechanics predicts a flat velocity curve.

As Jonathan said, it's wrong.
Your galaxy isn't spherical, you can't use your (wrong) formulas. Fix them and find the rotation curve of the bulge.
If you want to find the disk rotation curve you should write your potential considering a cylindrical distribution (hint: Green's functions), then
$$\frac{v_{c}^{2}}{R} = \frac{\partial \phi (R,z=0)}{\partial R}$$