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

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

The quantities to be discussed here are

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

For these quantities, [FORMULA], k, l, and [FORMULA] ares shown in Table 1. The right expression for [FORMULA] is valid if the velocity distribution does not depend on x, [FORMULA] denotes the semimajor axis in units of Einstein radii.


[TABLE]

Table 1. [FORMULA], k, l, and [FORMULA] for [FORMULA], [FORMULA], µ, T


For the galactic halo, a velocity distribution of

[EQUATION]

can be used, where [FORMULA].

The mass density of halo objects is modelled as

[EQUATION]

where r measures the distance from the Galactic center, [FORMULA] is the distance from the sun to the Galactic center, a is a characteristic core radius and [FORMULA] 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 [FORMULA], one obtains

[EQUATION]

where [FORMULA] is the angle between direction of the Galactic center and the direction of the LMC measured from the observer.

With

[EQUATION]

H(x) can be written as

[EQUATION]

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

[EQUATION]

and

[EQUATION]

Other values of [FORMULA] can be obtained numerically.

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

[EQUATION]


[TABLE]

Table 2. Selected values for [FORMULA]



[TABLE]

Table 3. Selected values for [FORMULA]



[TABLE]

Table 4. [FORMULA] for [FORMULA], [FORMULA], µ, T


From these values, one obtains for the expectation values:

[EQUATION]

With [FORMULA] one obtains

[EQUATION]

so that for [FORMULA], one gets [FORMULA] and [FORMULA].

The distribution of [FORMULA] is given by

[EQUATION]

and the probability density [FORMULA] is

[EQUATION]

From Eq. (93) the probability density [FORMULA] for the mass µ (in units of [FORMULA]) follows as

[EQUATION]

which differs from the probability given by Jetzer & Massó (1994 , Eq. (8)), by a factor [FORMULA]. Note that this probability density has to be of the form

[EQUATION]

since [FORMULA], to ensure normalization for any [FORMULA].

For the other quantities estimated the probability densities are given by Eqs. (60) and (61). The probability densities [FORMULA] and [FORMULA] for [FORMULA], µ, T are shown in Figs. 1 to  3. Note that [FORMULA] follows the velocity distribution, because [FORMULA]. In the diagrams for [FORMULA], symmetric intervals around [FORMULA] 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]

Table 5. The bounds of symmetric intervals on a logarithmic scale around [FORMULA] which correspond to probabilities of 68.3 % and 95.4 %


The bounds are much larger for [FORMULA] and again much larger for µ than for T, which is due to the wide distribution of the velocity and [FORMULA], [FORMULA], while [FORMULA]. The smallest and the largest value in the 95.4 %-interval differ by a factor of about 800 for µ, 16 for [FORMULA] and 5 for T.


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

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

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

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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