AST1420 - Lecture L3

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AST1420 - Lecture L3 Potential - density pairs
Newtons gravity Spherical systems - Newtons
theorems, Gauss theorem - simple sphercal
systems Plummer and others Flattened systems
- Plummer-Kuzmin - multipole expansion other
transform methods
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Below are large portions of Binney and Tremaine
textbooks Ch.2.
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This derivation will not be, but you must
understand the final result
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An easy proof of Newtons 1st theorem re-draw
the picture to highlight symmetry, conclude that
the angles theta 1 and 2 are equal, so masses of
pieces of the shell cut out by the beam are in
square relation to the distances r1 and r2. Add
two forces, obtain zero vector.
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This potential is per-unit-mass of the test
particle
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General solution. Works in all the spherical
systems!
Inner outer shells
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If you can, use the simpler eq. 2-23a for
computations
Potential in this formula must be normalized to
zero at infinity!
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(rising rotation curve)
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Another example of the use of Poisson eq in
the search for rho.
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Know the methods, dont memorize the details of
this potential-density pair
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Spatial density of light
Surface density of light on the sky
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Almost Keplerian
Linar, rising
Rotation curve
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An important central-symmetric potential-density
pair singular isothermal sphere
Do you know why?
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Empirical fact to which well return...
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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! Its not just the second
derivative. Constant b is known as the core
radius. Do you see that inside rb rho becomes
constant?
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Axisymmetric potential Kuzmin disk model
This is the so-called Kuzmin disk. Its
somewhat less useful than e.g., Plummer sphere,
but hey its a relatively simple potential -
density (or rather surface density) pair.
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Often used because of an appealingly flat
rotation curve v(R)--gt const at R--gt inf
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Caption on the next slide
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(Log-potential)
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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.
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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.
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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.
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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 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
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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, 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 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
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The second part of the lecture is a repetition of
the useful mathematical facts and the
presentation of several problems
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This problem is related to Problem 2.17 on p. 84
of the Sparke/Gallagher textbook.
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