Flattened systems
- Plummer-Kuzmin
- multipole expansion & other transform methods
There is nothing more practical than theory:
- Gauss theorem in action
- using v = sqrt(GM/r)
Very frequently used: spherically symmetric Plummer pot. (Plummer sphere)
Notice and
remember how the
div grad
(nabla squared or
Laplace operator in
eq. 2-48) is
expressed
as two consecutive
differentiations over
radius! It’s not just
the second
derivative.
Constant b is known
as the core radius.
Do you see that
inside r=b
rho becomes
constant?
Axisymmetric potential: Kuzmin disk model
This is the so-called
Kuzmin disk.
It’s somewhat less useful than
e.g., Plummer sphere, but
hey… it’s a relatively simple
potential - density (or rather
surface density) pair.
Often used because of an appealingly flat rotation curve v(R)--> const at R--> inf
Useful approx. to
galaxies if
flattening is small
Not very
useful approx. to
galaxies if
flattening not <<1,
i.e. q not close to 1
(Log-potential)
This is how the Poisson eq looks like
in cylindrical coord. (R,phi, theta) when nothing depends
on phi (axisymmetric
density).
Simplified Poisson eq.for very flat systems.
This equation was used in our Galaxy to estimate the
amount of material (the r.h.s.) in the solar neighborhood.
Poisson equation: Multipole expansion method.
This is an example
of a transform method:
instead of solving
Poisson equation in
the normal space
(x,y,z), we first
decompose density
into basis functions
(here called spherical
harmonics Yml) which
have corresponding
potentials of the same
spatial form as Yml, but
different coefficients.
Then we perform a
synthesis (addition)
of the full potential
from the individual
harmonics multiplied
by the coefficients
[square brackets]
We can do this since
Poisson eq. is linear.
In case of spherical harmonic analysis, we use the spherical coordinates.
This is dictated by the simplicity of solutions in case of spherically symmetric
stellar systems, where the harmonic analysis step is particularly simple.
However, it is even simpler to see the power of the transform method in the
case of distributions symmetric in Cartesian coordinates. An example will
clarify this.
Example: Find the potential of a 3-D plane density wave (sinusoidal perturbation
of density in x, with no dependence on y,z) of the form
 ( x , y , z )   ( k ) exp(i k  r )
k  wave vector  ( k x ,0,0)
 ( k )  front coefficient , const . w .r .t . r
r  position vector  ( x , y , z )
k  r  kx x
Euler ' s formula : exp(iq )  cos q  i sinq
We use complex variables (i is the imaginary unit) but remember that the
physical quantities are all real, therefore we keep in mind that we need to drop
the imaginary part of the final answer of any calculation. Alternatively, and more
mathematically correctly, we should assume that when we write any physical
observable quantity as a complex number, a complex conjugate number is
added but not displayed, so that the total of the two is the physical, real number
(complex conjugate is has the same real part and an opposite sign of the
imaginary part.) You can do it yourself, replacing all exp(i…) with cos(…).
Before we substitute the above density into the Poisson equation, we assume
that the potential can also be written in a similar form
( x , y, z )  ( k ) exp(i k  r )
Now, substitution into the Poisson equation gives
 2   4 G 
( k )  2 exp(i k  r )  4 G 
 k 2 ( k ) exp(i k  r )  4 G   4 G  ( k ) exp(i k  r )
4 G  ( k )
( k )  
k2
where k = kx, or the wavenumber of our density wave. We thus obtained a very
simple, algebraic dependence of the front coefficients (constant in terms of x,y,z, but
in general depending on the k-vector) of the density and the potential. In other words,
whereas the Poisson equation in the normal space involves integration (and that can
be nasty sometimes), we solved the Poisson equation in k-space very easily. Multiplying
the above equation by exp(…) we get the final answer 
4 G  ( x , y , z )
( x , y , z )  
k2

As was to be expected, maxima (wave crests) of the 3-D sinusoidal density wave
correspond to the minima (wave troughs, wells) of its gravitational potential.
x
The second part of the lecture is a repetition of the
useful mathematical facts and the presentation of
several problems
This problem is related to Problem 2.17 on p. 84 of the Sparke/Gallagher textbook.