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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

3 Answers 3

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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This is answering a completely different question to the one posed. It is a good answer to that question (i.e why isn't dark matter dynamically important in te solar system?) but I'm baffled that it has attracted upvotes for this question, which is actually about the Galactic rotation curve interior to the Sun's orbit. – Rob Jeffries Mar 8 at 18:18

This is not my field, but I think it is as simple as a combination of the things you said - accurately estimating the amount of "normal matter" within the solar radius, accurately measuring the rotation curve (different tracers can give slightly different results), but mainly that dark matter does not dominate the dynamics at small radii.

For instance we know what the mass density is in the local disk by looking at the velocity dispersion of stars perpendicular to the plane (e.g. Kuijken & Gilmore 1989; Creze et al. 1998). These results show that dark matter is hardly present in the disk at the radius of the Sun - the local mass density is entirely accounted for by stars, gas and stellar remnants.

There is no real reason why the (dissipationless) dark matter should be as centrally concentrated as the luminous matter, and that appears to be the case.

Models for the Milky Way dark halo suggest it is distributed with far less "concentration" than the mass in the disk. The scale length parameter of the Navarro-Frenk-White profile is typically found to be $\sim 20$ kpc (e.g. Klypin et al. 2001), whereas the exponential scale length of the disk is more like 3-4 kpc. The plot below shows how various components contribute to the rotation curve. See how dark matter is only dynamically important at large radii, beyond the solar orbit. So I think that even if you have very good measurements it is still difficult to get a handle on the dark matter profile from the rotation curves because it doesn't change the overall rotation curve prediction that much at small radii.

Contributions to the rotation curve

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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:

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