**2.4. Matching Pseudo-Isothermal Models to the Isothermal Sphere**

In carrying out rotation curve decompositions, some authors model the
DM with a nonsingular isothermal sphere; then we adopt the parameters
*r*_{c},
_{0},
and without
modification. Other authors use the
"pseudo-isothermal sphere" (hereafter the PITS), i.e., the
approximation that the volume density is

(3) |

They then derive parameters
_{0},
*a*, and the asymptotic circular velocity
*V*() as
*r*
. No compelling physical
argument favors one model over the other. However, if we want to combine
data from different sources, we need to correct parameters determined
using the PITS to the ones that would have have been measured using the
isothermal. An exact correction is not possible, because equation (3) is
a poor approximation to the isothermal sphere except as
*r*
.
Figure 1 shows three possible scalings of the
rotation and velocity dispersion curves of the PITS to the
isothermal. All three scalings are shown to emphasize the physical
difference between the two models. The scalings are not equally
plausible, but each has a simple underlying motivation. We will find
that the best scaling is intermediate between the middle and right
panels of Figure 1.

In all three scalings, the central densities of the two models are the same. The panels differ in how radii and velocities are scaled.

At left, the PITS is scaled in *r* and *V* so that the central
densities and asymptotic circular velocities are the same. Then the core
radius *r*_{c} of the isothermal is
*r*_{c} = 3*a* / 2^{1/2}. The dispersion scale
in the top panel is the velocity dispersion of the isothermal sphere,
but this is equal to the local dispersion of the PITS only as
*r*
. For most
galaxies, the HI distribution does not extend beyond
~ 2.5 *r*_{c}.
Here the *local* dispersion of the PITS is less than that of the
isothermal (Fig. 1, *top left*). The reason
is that an isotropic system with the density distribution of equation
(3) is not isothermal; its velocity dispersion is

(4) |

where *x* = *r* / *a*. Since the analytic model matches
the isothermal badly in both *V*(*r*) and
(*r*), we discard
this scaling.

Given the limited extent of HI data, it seems more realistic
to scale the PITS so that its rotation curve is similar to that
of the isothermal at
*r* 2.5
*r*_{c}. We get such a scaling
(Fig. 1, *right*) if we solve our problem
with the halo dispersion by making the velocity dispersion of the
corresponding isothermal be equal to the *central* dispersion
(0) =
(^{2} / 8 -
1)^{1/2}
*V*() = 0.4834
*V*()
of the PITS. Then *r*_{c} = (0.6837)3*a* /
2^{1/2} = 1.4503 *a* and
= 0.4834
*V*(). The
resulting parameters are related in the
normal way for an isothermal sphere,
^{2} =
4
*G*
_{0}
*r*_{c}^{2} / 9.

The virtue of this scaling is that the two halo rotation curves are
similar over the radius range in which they are actually fitted to the data
(Fig. 1, *bottom right*). However, there is
a problem (Fig. 1, *top right*). This
scaling forces
*V*() to be much
larger than the
velocity of the flat part of the rotation curve. An examination of published
decompositions shows that authors almost never extrapolate implicitly to
such a large maximum rotation velocity - one that is never observed. The
exceptions are some rotation curve decompositions in which
*V*_{DM}
*r* over the
whole radius range of the data, i.e., cases in which *r*_{c} is
completely unconstrained. Examples are the decompositions of
NGC 7331 and NGC 6674 in
Broeils (1992).
We discard these decompositions anyway.

This suggests a compromise scaling like the one in the middle panel. Here
*V*() is scaled to
the *maximum*, not the asymptotic, rotation
velocity of the isothermal. Then, when authors use equation (3) to derive
_{0},
*a*, and *V*()
via rotation curve decomposition, we would adopt
*r*_{c} = 1.8918 *a*, the authors' quoted value of
_{0},
and a halo velocity dispersion
= 0.6306
*V*(). Again, these
parameters are related as normal for an isothermal sphere.

In practice, there is no guarantee that any of the above scalings
represents what happens when one author uses the PITS and another uses the
isothermal in rotation curve decomposition. We note again that
Figure 1 is
included to emphasize that the PITS is not very isothermal. However, we will
find in Section 3 that enough
decompositions have been published using
each of the above models so that we can derive the DM parameter correlations
separately for each model. We will then derive the best scaling of one
model to
the other by matching their respective DM correlations. This scaling is used
to construct Figure 4, which
combines data from all sources. It is intermediate between the middle
and right-hand panels of Figure 1, i.e.,
_{0} =
0.9255 _{0,
PITS}; *r*_{c} = 1.6154 *a*, and
= 0.7334
_{PITS} = 0.5186
*V*_{, PITS}.