How does a dark matter halo outside a galaxy help to explain a galaxy rotation curve?
Suppose for simplicity we use a model of a star rotating about a more massive star in a fixed circular orbit. For another star at larger radius, the velocity should be smaller but observationally for spiral galaxies, the opposite happens (moves at even more speed or at least at the same speed as the closer star). People hypothesize that dark matter halo may be present “surrounding” the system (i.e., halo exists outside the system) to account for such observation. I feel that exactly the opposite should be the case that is, dark matter halo should be inside the galaxy rather than outside. Reason being, if dark matter is outside the galaxy then the effective gravitational pull on the rotating star towards the center will be reduced due to outward attraction on the star exerted by dark matter existing outside. Thus, its speed of rotation should be reduced; but we observe the opposite. So how does dark matter halo "surrounding" a galaxy help to explain galaxy rotation curve? Where am I making mistake in argument?
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