Sean said:
I think the calculations and observations show that the main force in our solar system is our sun and that all the dark matter in our solar system does not constitute a mass greater than our sun where as dark matter is more dense towards the centre of our galaxy creating a force upon the stars.This is how I am seeing it let me know if that's wrong.And if it is wrong then my question still stands.Or is it that we just live in an area where dark matter is scarce.
No, The increasing density towards the center is not relevant here.
What determines how an object orbits is the total mass inside the spherical volume which has a radius equal to the distance of the object from the center. For a planet in the Solar system, the vast majority of the mass inside this "interior sphere" resides in the Sun, so as you move outward from planet to planet, it stays almost the same. This is why the planets orbit the Sun with the velocities they have.
If we look at a galaxy, its is a bit different. At the center you are in the central bulge, and as you move away from the center, the mass inside your interior sphere grows rapidly, so fast that the its increase overrides the effect of your increasing distance and your orbital speed increases as you move outward.
Once you leave the bulge and enter the region of the disk, the mass increase with distance should drop off sharply (as long as you only consider the visible matter), and your orbital velocity should begin to decrease as you move outward.( similar to how it does with the solar system). It doesn't, and that is where dark matter comes in.
Dark matter is not distributed in the same was as the stars making up the visible galaxy are. It continues to spread out spherically from the center even once you have left the central bulge. So as you move outward the mass it contributes to determining your orbit continues to increase rapidly. This means that when you get out into the disk, Your interior sphere encompasses a large volume which is taken up in part by visible matter, but mostly consists of space filled with just dark matter. Since this is a very large volume, the dark matter doesn't have to be very dense to add up to a considerable amount of mass.
To give you an idea of what I mean, consider this: Even if you assumed that dark matter was evenly distributed at the same density as it is found in the Solar system, the total amount of the mass of dark matter, spread out in the spherical volume with a radius equal to the distance of our Solar system from the center of the galaxy, would add up to a large fraction of the estimated total mass of the visible matter in the galaxy.
In other words, Dark matter doesn't be of a great density to have a profound effect because of the huge volume that it occupies.