The Distributions of the Density and the Mass for Spiral Galaxies

Expressions for the distributions of the density and mass of spiral galaxies have been derived for three different orientations of rotation curves with a turn over radius (rt). The central region of the galaxy is found to rotate as a rigid body with a constant density and with a central mass as a function of r. The density of the outer regions (r > rt) is found to vary inversely with the radius (r), and the corresponding mass is found to be a function of r. Galaxies with flat rotation curves at (r > rt) has less mass than that with rising up or lowering down rotation curves.


INTRODUCTION
The rotation of spiral galaxies was discovered by Slipher (1914) when he detected inclined absorption lines in the spectrum of M31 galaxy.
Rotation curves of spiral galaxies are one of the main tools to study the distribution of mass in spiral galaxies.They constitute the best observational proof for the existence of dark matter in spiral galaxies.Rotation curves provide important information for understanding the dynamics, evolution and formation of galaxies.
Rotation curves which are useful to derive the mass distribution in spiral galaxies are obtained by observing the emission lines of ionized gas at optical frequencies such as Hα or the 21 cm line of neutral hydrogen at radio frequencies (wright, 1971;Roberts and Rots, 1973).
The main tool that we had used for this study is the rotation curves of spiral galaxies.Although its already 50 years that the rotation curves are used in order to derive the matter distribution (density law of galaxies and galactic mass) in spirals, a different approaches can be always applied to get a more relevant information.For this purpose we started with what is called a density law for spiral galaxies derived by Daoud et al., 2009, which is a solution of a general Burbidge equation related the galactic velocity to the galactic density (Binney and Tremaine, 1987).
In this paper we studied the galactic density and mass of spirals that their rotation curves show a turn over at some distance from the center of the galaxy (radius) r t , like NGC 4378, NGC 4594, NGC 3145, NGC 7664 and others Fig.

(1).
At radii less than the turn over radius (r < r t ) it has been found that the behavior of the density and the mass for all galaxies are the same.i.e. the central density is found to be a constant while the central mass turns to be a function of r 3 .
At radii larger than the turn over radius (r < r t ), the rotation curves of spiral galaxies show three different cases.In case one the curve goes up slightly as in NGC 4594, in case two the curve is almost flat as in NGC 3145), while in case three, the curve goes down slightly (NGC 4378).In all the three cases mentioned above the galactic density at (r > r t ) is inversely proportional to the radius (r), therefore, the density decreases for large radii.In case 2 where the curve is flat, we found that the galactic mass for the outer regions of the galaxy is very small comparing with the other two cases, while the mass of the outer regions in case 3 is the largest.

CALCULATIONS AND RESULTS
To study the density of spiral galaxies we started with the following equation: from (Daoud et al., 2009), where ρ is the galactic density, k is the eccentricity of the system and V(a) is the velocity at distance a from the center of the galaxy.
Rotation curves of spiral galaxies with a turn over radius take different orientations at (r > r t ) as follows.
Case 1 [The curve abc 1 ]: Some spiral galaxies show rotation curves (abc 1 ) which rise up slightly at (r > r t ) (the curve bc1 in Fig. 2).
In this case rotation curves can be approximated by two linear equations.
( ) Where α is a constant which refers to the slope of the inner part of the rotation curve (the slope of the line ab Fig. 2).and Where β is a constant which refers to the slope of the outer part of the rotation curve (the slope of the line bc 1 in Fig. 2), and γ is another constant which can be found from the intersection of the outer part of the rotation curve (bc 1 ) with the vertical axis.
Let ρ 1 (r,k) be the density of the inner region of the galaxy (r < r t ) where V(a) = αa.
Substituting equation (2) in equation ( 1) and letting k r x = we have.
( ) And then differentiating we get.
Equation ( 6) shows that the density of the inner parts of the galaxy is in depended of r (ρ is constant), this means that the central part of the galaxy rotates like a rigid body which is consistence with the texts (Mihalas and Binney, 1968 ;Carroll and Ostlie, 1996).
To find an expression for the mass of the galaxy, we use the following equation.
Since we divided the rotation curve into two regions then let us first find the inner mass of the galaxy m 1 (r), with the central density ρ 1 (r, k) at (r < r t ).
Therefore the inner mass of the galaxy at (r < r t ) is proportional to r 3 , in other words the inner mass of the galaxy increases sharply as we go farther from the galactic center.
To find the density at (r > r t ), we substitute equation (3) in equation ( 1), letting k r x = and using the following standard integrals (William and Beyyer, 1985).
ρ 2 (r, k) should be the density for the outer region of the galaxy (from r t to r).According to equation (1) the limits of integration cannot be changed, for this reason, in calculating ρ 2 (r, k) we integrated from zero (at which V(r) = γ) to r (the edge of the galaxy) and then we found the outer density of the from.
Therefore the outer density of the galaxy, ρ 3 (r, k), is proportional to r -1 i.e. the halo density decreases as we go farther from the turn over radius.This result is consistent with the result of Carrol and Ostlie (1996).
The outer mass of the galaxy (at r > r t ) is evaluated by combining equations 7 and 13, then 14) Equation ( 14) indicate that the mass of the outer regions of the galaxy is proportional to r 2 .
The inner mass of any spiral galaxy of this type of rotation curve can be calculated from equation ( 9) by letting r = r t and the mass of the outer regions of the galaxy can be calculated from equation ( 14) by letting r = r g where r g is the radius of the given galaxy, therefore the total mass of the galaxy is given by: ( ) ( ) To find the numerical value of the galactic mass (equation 14), the constants α, β, γ can be taken from the velocity curve, the eccentricity k can be taken from observations and r t can be approximated from the velocity curve or it can be found by equating equations ( 6) and (11) so it turns to be Case 2 (The curve abc 2 ): For the second case, the rotation curves of some spiral galaxies are almost flat (β→0) at r > r t (the curve bc 2 , Fig. 2).In this case we don't have to go through all derivations again, but all we have to do is setting β very small (β→0) in equation ( 13) and ( 14).In equation (13) if β is very small then ρ 3 (r, k) is small too as well as m 2 (r).In other words when the second part of the rotation curve is almost flat then the mass of the outer parts of the galaxy m 2 (r) drops down quickly comparing with the inner mass, therefore, the contribution of the halo mass to the total mass of the galaxy is small comparing with case 1.

Fig. 2 :
Fig.2: Three general forms of rotation curves for spiral galaxies