Dark matter density calculation

[skydivephil]

Homework Statement

We have been asked to calculate the density of dark matter in the solar system. I've used the following equation:

Homework Equations

rho= 3v^2/4gPir^2

The Attempt at a Solution

V = 2.2*10^5 speed of solar system orbit around the galaxy
G = 6.67*10^-11 gravitational constant
PI 3.145
R = 2.5*10^20 the radius of the suns distance to the galactic centre in meters
SO 3*2.2*10^5= 1.45*10^11
Divided by
4*6.67*10^-11*3,14*8.95*10^41
=2.77*10^-21
Wen my lecturer gave us this equation he didnt specify what units the answer should be.
Assuming its kg/m^3
I then tried to work out the mass of the solar system , via assuming
mass = density*volume.
I assumed i should get a number about 5 times the mass of the sun as dark matter normally outweighs baryonic matter by about this ratio.
Instead
I got Density is equal to mass over volume
P = m/v
We know know
P = 2.77*10^21 kg/m^3
The volume of a sphere is 4/3Pir^3
We need to know the radius of the solar system, this is not an easy to number to determine as there is no definite edge to the soalr system. However 100 Au is usually quoted.
We have to covert this to meters. 1 Au is 1.49*10^11
100 times this is 1.49*10^13
4/3*Pi*(1.49*10^13^3)
= 1.4*10^40 m^3
No we can determine the mass of the solar system
Reaaranging the equation
P=m/v to solve for for m
We have
M= p*v
2.77*10^-212kg m^3*1.4*10^40
=3.8*10^19
which is less than the mass of the sun! the mass of the sun is 2*10^30kg

Where have i gone wrong?

 

[Redbelly98]

Homework Statement

We have been asked to calculate the density of dark matter in the solar system. I've used the following equation:

Homework Equations

rho= 3v^2/4gPir^2

The Attempt at a Solution

V = 2.2*10^5 speed of solar system orbit around the galaxy
G = 6.67*10^-11 gravitational constant
PI 3.145
R = 2.5*10^20 the radius of the suns distance to the galactic centre in meters
SO 3*2.2*10^5= 1.45*10^11
Divided by
4*6.67*10^-11*3,14*8.95*10^41
=2.77*10^-21

Looks about right to me.

Wen my lecturer gave us this equation he didnt specify what units the answer should be.
Assuming its kg/m^3

I've pretty much seen two approaches students take to handling units. One is to simply plug in numbers, and assume the units work out to whatever they should be. But if you have doubts, you try the second method instead: put in whatever units you have for v, G, and r, and then see what the units come out to be for v²/(Gr²). Note that the 3/(4pi) do not contribute to the units.

I then tried to work out the mass of the solar system , via assuming
mass = density*volume.

Is this something you are doing for your own benefit, i.e. it was not given in the problem statement you wrote.

I assumed i should get a number about 5 times the mass of the sun as dark matter normally outweighs baryonic matter by about this ratio.

That's on average throughout the galaxy. Dark matter is not concentrated within solar systems, it is spread out between the stars as well. But it is good that you are thinking up front about what sort of answer to expect.

Instead
I got Density is equal to mass over volume
P = m/v
We know know
P = 2.77*10^21 kg/m^3
The volume of a sphere is 4/3Pir^3
We need to know the radius of the solar system, this is not an easy to number to determine as there is no definite edge to the soalr system. However 100 Au is usually quoted.
We have to covert this to meters. 1 Au is 1.49*10^11
100 times this is 1.49*10^13
4/3*Pi*(1.49*10^13^3)
= 1.4*10^40 m^3
No we can determine the mass of the solar system
Reaaranging the equation
P=m/v to solve for for m
We have
M= p*v
2.77*10^-212kg m^3*1.4*10^40
=3.8*10^19
which is less than the mass of the sun! the mass of the sun is 2*10^30kg

Where have i gone wrong?

Since dark matter is spread out between stars, and not concentrated within the solar system, a better volume estimate would be a cube or sphere that takes into account the average spacing between stars in our neighborhood of the galaxy.

Also, remember that using just the mass of the sun and Newton's gravity formula gives a pretty accurate calculation for the orbits of the planets. The effects of dark matter were not noticed until people looked on a galactic, not solar system, scale.

 

[skydivephil]

Looks about right to me.



Is this something you are doing for your own benefit, i.e. it was not given in the problem statement you wrote.

That's on average throughout the galaxy. Dark matter is not concentrated within solar systems, it is spread out between the stars as well. But it is good that you are thinking up front about what sort of answer to expect.

Since dark matter is spread out between stars, and not concentrated within the solar system, a better volume estimate would be a cube or sphere that takes into account the average spacing between stars in our neighborhood of the galaxy.

Also, remember that using just the mass of the sun and Newton's gravity formula gives a pretty accurate calculation for the orbits of the planets. The effects of dark matter were not noticed until people looked on a galactic, not solar system, scale.

Thanks a lot for you reply.

Yes thhe question also asked us to compare the the mass of dark
matter within the Solar System with the mass of normal matter?
If iVe applied everything correctly then the answer I got for

rho= 3v^2/4gPir^2
=2.77*10^-21
gave us a mass of =3.8*10^19 Kg
Given the suns mass is 2*10^30kg
This impleis baryonic matter outnumbers dark matter by 5.26*10^10.
I have to say I assumed I had to have done something wrong given dark matter outnumbers baryonic matter. But I take the point about it being undetectable in the solar system due to the ability of Kelpers laws to predict planetary motions. So you think these numbers look right?

 

[Redbelly98]

Yes, your numbers look reasonably given the 100 a.u. radius, which as you said is uncertain.