Date post: | 04-Jan-2016 |
Category: |
Documents |
Upload: | brandon-knight |
View: | 218 times |
Download: | 1 times |
Newton’s gravity
Spherical systems - Newtons theorems, Gauss theorem - simple sphercal systems: Plummer and others
Flattened systems - Plummer-Kuzmin - multipole expansion & other transform methods
[Below are large portions of Binney and Tremaine textbook’s Ch.2.]
This derivationwill not be,but you must understand thefinal result
An easy proof of Newton’s 1st theorem:re-draw the picture to highlight symmetry,conclude that the angles theta1 and 2 are equal, so masses of pieces of the shell cut out by thebeam are in square relation to the distances r1 and r2. Add two forces, obtain zero vector.
This potential is per-unit-mass
of the test particle
General solution. Works in all the spherical systems!
Inner & outer shells
If you can, use the simplereq. 2-23a for computations
Potential in this formula must benormalized to zero at infinity!
(rising rotation curve)
Another exampleof the use of Poisson eq in thesearch for rho.
Know the methods, don’t memorize the details of this potential-density pair:
Spatial density of light
Surface density of lighton the sky
Rotation curve
Linar, rising
Almost Keplerian
Do you knowwhy?
An important central-symmetric potential-density pair: singular isothermal sphere
Empirical factto which we’ll return...
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
Caption on the next slide
(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.