SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 342, 34-40 (1999)

Previous Section Next Section Title Page Table of Contents

Appendix A: derivation of [FORMULA]

In order to find [FORMULA] we have to calculate the integral in Eq. (9) (see Sect. 2). We change the integration variable in Eq. (9) from [FORMULA] to [FORMULA] using the following relations:

[EQUATION]

[EQUATION]

we have that Eq. (9) can be rewritten as:

[EQUATION]

and if we express the product [FORMULA] as a function of [FORMULA] (CN43 - Eq. 18) we can write [FORMULA] in the following form:

[EQUATION]

where [FORMULA] is given by:

[EQUATION]

or equivalently:

[EQUATION]

The first integral in Eq. (A6) can be easily calculated:

[EQUATION]

with

[EQUATION]

For [FORMULA] Eq. (A7) gives:

[EQUATION]

that coincides with the first term in the right hand side of Eq. (22) in CN43.

The second integral is more difficult to evaluate and is given by:

[EQUATION]

We have obtained this last result using Eq. (10) and Eq. (12) (see Sect. 2). Since we are interested in the moments of [FORMULA] for a given [FORMULA] we need only the behaviour of the function [FORMULA] for [FORMULA]. To obtain this we expand the term [FORMULA] in powers of [FORMULA] and [FORMULA] in Eq. (A10) and then we have:

[EQUATION]

or

[EQUATION]

where [FORMULA] is given by

[EQUATION]

being [FORMULA]. Substituting these last equations in Eq. (A12) we have:

[EQUATION]

Now, we evalutate this integral following CN43. We introduce a system of coordinates with the z-axis in the direction of [FORMULA] and letting [FORMULA], [FORMULA] and [FORMULA]. The result of this integration is given by:

[EQUATION]

where

[EQUATION]

For [FORMULA] Eq. (A15) becomes

[EQUATION]

which coincides with Eq. (37) in CN43.

The second integral in Eq. (A12) is:

[EQUATION]

If we define the variable

[EQUATION]

and we introduce Eq. (A13) in Eq. (A18) we obtain:

[EQUATION]

where [FORMULA] and w is the azimuthal angle. The integral can be calculated regarding z and t as complex variables and using an appropriate chosen contour (see Chandrasekhar & von Newmann 1942). The result of the integration in terms of the original variable [FORMULA] is:

[EQUATION]

where

[EQUATION]

then we have obtained:

[EQUATION]

Now substituting Eq. (A28) in Eq. (A4) we have:

[EQUATION]

where we have used bars to indicate that the corresponding quantities have been averaged with the weight function [FORMULA].

If the distribution of the velocities of the field stars, [FORMULA], is spherical and the test star moves with velocity [FORMULA] we have:

[EQUATION]

and also:

[EQUATION]

[EQUATION]

[EQUATION]

and if we use the system of coordinates introduced by CN43 that is [FORMULA], [FORMULA] and [FORMULA] where [FORMULA] is the angle between [FORMULA] and [FORMULA] we can simplify Eq. (A29) and then the Eq. (8) (see Sect. 2) becomes:

[EQUATION]

where [FORMULA]. We define, now, the following constants and functions:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

so we can re-write Eq. (A34) as:

[EQUATION]

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
helpdesk.link@springer.de