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It seems that we know the rotation curve inside the sun's galactic orbit fairly accurately. Then wouldn't we be able to just take the derivative* of this to get the DM density profile at smaller radii? What then is the primary reason we can't distinguish between different profiles like pseudo-isothermal, NFW etc?

a) Is it uncertainty in the rotation curve? If so what is the source of this uncertainty? b) Uncertainty in the amount of luminous matter? Source?

Something else?

I can see why the profile is uncertain at larger distances from the galactic center since the rotation curve is uncertain there, but why is it uncertain in the region where the rotation curve is known more precisely?

*Well, square, multiply by r and take the derivative, but you get the idea

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Not my field, but I suspect the answer lies in the ratio of the volume of the solar system to the volume of galactic space in which the sun is the gravitational dominant mass. There is a lot of dark matter, but it is spread over a whole lot of space. Cue the usual Douglas Adams quote. –  dmckee Mar 7 '13 at 2:24

2 Answers 2

The average dark matter density in the universe is about the mass of 1 hydrogen atom per cubic metre. If we take this density we can work out how the mass of the dark matter within e.g. the orbit of Jupiter compares to the mass of the Sun.

The distance of Jupiter from the Sun is about 800,000,000,000 metres (I say about because it's orbit is elliptical), so the mass of dark matter within it's orbit is 2 $\times$ 10$^{36}$ hydrogen atoms or about 3.5 $\times$ 10$^9$ kg. This is about 2 $\times$ 10$^{-19}$% of the mass of the Sun.

We might expect the density of dark matter in the Solar system to be higher than the average over the whole universe, but not by anything approach a factor of 10$^{21}$. On the scale of the Solar System the effect of the dark matter is entirely negligable. It's only when you get to interstellar distances that it starts becoming significant.

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We know the relative velocity w.r.t. to the Sun inside the solar radius, and could get from there the rotation curve if we knew the rotation velocity of the Sun (or of the Local Standard of Rest) in the Galaxy, and the distance of the Sun from the Galactic center. We dont know that distance very precisely ($8\,\text{kpc} \pm 1.5\,\text{kpc}$), and we dont know the velocity of the Sun w.r.t. to the Galactic center at all ($230 \,\frac{\text{km}}{\text{s}} \pm 50 \,\frac{\text{km}}{\text{s}}$).

We do know however that the ratio of these two quantities is about $30 \,\frac{\text{km}}{\text{s}\cdot \text{kpc}}$. See McMillan and Binney for all details: http://arxiv.org/abs/0907.4685

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