8. Estimates and probability distributions for physical quantities for a simple galactic halo model
For these quantities, , k, l, and ares shown in Table 1. The right expression for is valid if the velocity distribution does not depend on x, denotes the semimajor axis in units of Einstein radii.
Table 1. , k, l, and for , , µ, T
For the galactic halo, a velocity distribution of
can be used, where .
The mass density of halo objects is modelled as
where r measures the distance from the Galactic center, is the distance from the sun to the Galactic center, a is a characteristic core radius and is the local density at the position of the sun.
With the distance parameter x, which measures the distance along the line-of-sight from the observer to the LMC in units of the total distance , one obtains
where is the angle between direction of the Galactic center and the direction of the LMC measured from the observer.
H(x) can be written as
Let the halo be extended to a distance of along the line-of-sight. With and , one obtains
Other values of can be obtained numerically.
Table 2. Selected values for
Table 3. Selected values for
Table 4. for , , µ, T
With one obtains
so that for , one gets and .
and the probability density is
From Eq. (93) the probability density for the mass µ (in units of ) follows as
which differs from the probability given by Jetzer & Massó (1994 , Eq. (8)), by a factor . Note that this probability density has to be of the form
since , to ensure normalization for any .
For the other quantities estimated the probability densities are given by Eqs. (60) and (61). The probability densities and for , µ, T are shown in Figs. 1 to 3. Note that follows the velocity distribution, because . In the diagrams for , symmetric intervals around are shown which give a probability of 68.3 % and 95.4 % respectively. The bounds of these intervals are also shown in Table 5.
Table 5. The bounds of symmetric intervals on a logarithmic scale around which correspond to probabilities of 68.3 % and 95.4 %
The bounds are much larger for and again much larger for µ than for T, which is due to the wide distribution of the velocity and , , while . The smallest and the largest value in the 95.4 %-interval differ by a factor of about 800 for µ, 16 for and 5 for T.
© European Southern Observatory (ESO) 1998
Online publication: January 27, 1998