Astron. Astrophys. 330, 963-974 (1998)

8. Estimates and probability distributions for physical quantities for a simple galactic halo model

The quantities to be discussed here are

• the transverse velocity ,
• the Einstein radius (which immediately yields the projected distance ),
• the mass ,
• the rotation period T of a binary.

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.

With

H(x) can be written as

Let the halo be extended to a distance of along the line-of-sight. With and , one obtains

and

Other values of can be obtained numerically.

Selected values for and and are shown in Tables 2, 3, and  4, using the values used by Paczynski (1986 ) ²

Table 2. Selected values for

Table 3. Selected values for

Table 4. for , , µ, T

From these values, one obtains for the expectation values:

With one obtains

so that for , one gets and .

The distribution of is given by

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.

Fig. 1. The probability density for (left) and The probability density with symmetric 68.3 % and 95.4 % intervals around or (right)

Fig. 2. The probability density for (left) and the probability density with symmetric 68.3 % and 95.4 % intervals around (right)

Fig. 3. The probability density for (left) and the probability density with symmetric 68.3 % and 95.4 % intervals around (right)

© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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