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)

Notice and remember how the div grad(nabla squared or Laplace operator in eq. 2-48) is expressed as two consecutivedifferentiations 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?

Very frequently used: spherically symmetric Plummer pot. (Plummer sphere)

This is the so-called Kuzmin disk. It’s somewhat less useful than e.g., Plummer sphere, buthey… it’s a relatively simplepotential - density (or rather surface density) pair.

Axisymmetric potential: Kuzmin disk model

Often used because of an appealingly flat rotation curve v(R)--> const at R--> inf

Useful approx. to galaxies ifflattening is small

Not very useful approx. to galaxies ifflattening 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 theamount 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 densityinto basis functions

(here called sphericalharmonics Yml) whichhave 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 sincePoisson 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 symmetricstellar systems, where the harmonic analysis step is particularly simple.

However, it is even simpler to see the power of the transform method in thecase 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

We use complex variables (i is the imaginary unit) but remember that thephysical quantities are all real, therefore we keep in mind that we need to dropthe imaginary part of the final answer of any calculation. Alternatively, and moremathematically correctly, we should assume that when we write any physicalobservable quantity as a complex number, a complex conjugate number isadded 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 assumethat the potential can also be written in a similar form

qiqqiformulasEuler

xkrk

zyxvectorpositionr

rtrwconsttcoefficienfrontk

kvectorwavek

rkikzyx

x

x

sincos)exp(:'

),,(

....,)(

),,(

)exp()(),,(

00

)exp()(),,( rkikzyx

Now, substitution into the Poisson equation gives

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, butin 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 canbe nasty sometimes), we solved the Poisson equation in k-space very easily. Multiplyingthe above equation by exp(…) we get the final answer

As was to be expected, maxima (wave crests) of the 3-D sinusoidal density wavecorrespond to the minima (wave troughs, wells) of its gravitational potential.

2

2

2

2

4

44

4

4

kkG

k

rkikGGrkikk

Grkik

G

)()(

)exp()()exp()(

)exp()(

x

2

4k

zyxGzyx

),,(),,(

The second part of the lecture is a repetition of theuseful mathematical facts and the presentation of several problems

This problem is related to Problem 2.17 on p. 84 of the Sparke/Gallagher textbook.