Thursday, November 7, 2013

Math Drives the Universe.......And Astronomy (Part 3)

By his junior and senior year, the Astronomy major who's survived thus far has much more math to look forward to, in courses such as: Introductory Astrophysics, Celestial Mechanics, Radio Astronomy, Stellar Constitution & Evolution, and Stellar Spectroscopy. In addition he will be expected to complete an undergrad research project, and attend Astronomy seminars in the department - in which visiting 'stars' present their latest research findings.

In Introductory Astrophysics, the primary emphasis is getting the student acquainted with the Planck function and applying it to simple, plane-parallel stellar atmospheres, such as depicted below:

The Planck function describes the distribution of radiation for a black body, and can be expressed:

B(l) = {(2 hc2)/ l5}  [1/ exp (hc/lkT) - 1)]

Of course, all stars are effectively black bodies, which are perfect radiators, or as close to that state as nature allows.  In the case of simple radiation transfer in a static model stellar atmosphere (e.g. nothing changes with time), we have the relation of specific radiation intensity I(l) to source function S(l):

dI(l)/ds = -k(l)  I(l) + k(l)  S(l)

 = k(l) [S(l) – I(l)] - 0   or I(l) = S(l)

For a black body, the student learns I(l) equals the Planck function:

B(l) :  i.e.  S(l) = I(l) = B(l)

And this is a condition which implies LOCAL THERMODYNAMIC EQUILIBRIUM or LTE   LTE does NOT mean complete thermodynamic equilibrium!(E.g. since in the outer layers of a star there is always large energy loss from the stellar surface) . Thus, one only assumes the emission of the radiation is the same as for a gas in thermodynamic equilibrium at a temperature (T) corresponding to the temperature of the layer under consideration.  Another way to say this is that if LTE holds, the photons always emerge at all wavelengths.  In the above treatment, note that the absorption coefficient was always written as: k(l) to emphasize its wavelength (l) dependence. 

The student will be required to do a number of challenging problems for homework, including the specific applications of the gray atmosphere approximation. In a particular integral, let the surface flux: p( Fo ) = 2 p (I(cos (q)) = p [a(l) + 2(b(l)/3 ] 

and Flo = S(l) t(l) = 2/3 

which states that the flux coming out  of the  stellar surface is equal to the source function at the optical depth t = 2/3. This is the very important ‘Eddington-Barbier’ relation that facilitates an understanding of how stellar spectra are formed.  Once one then assumes LTE, one can further assume k(l) is independent of l (gray atmosphere) so that: 

k(l) = k;  t (l) = t  and Flo =  Bl (T(t = 2/3) )

Thus, the energy distribution of Fl is that of a black body corresponding to the temperature at an optical depth t = 2/3.  From this, along with some simple substitutions and integrations a wdie array of problems can be done. A few HW problem  examples:

1. Estimate the specific intensity I (q=p/4) if the surface flux from the Sun is  6.3 x 10 7 Jm-2 s-1.

2.  Find the effective temperature of the Sun and the boundary temperature (To) and account for any difference. (Hint: The effective temperature is related to the boundary temperature by: Teff  = (2)1/4 To   )

3. The lines of the Balmer series crowd close together as higher series members are considered. If each line was exactly one A (1 Angstrom) wide, how many Balmer lines would be individually visible without overlapping other lines?

Celestial Mechanics:

The name conveys exactly what the subject embodies: Mechanics applied to the dynamics of celestial objects. A typical diagram from my own Celestial Mechanics notes when I took it at Univ. of South Florida, is shown below:

The diagram shows assorted "orbital elements" for a planet of mass m2 (the Sun is m1) and this can be used as a basis for orbital energy analysis and also to predict future positions. Energy constants in celestial mechanics are very useful for quickly coming to terms with specific properties of an orbit such as shown in the accompanying sketch- designating a generic orbit in x-y-z space. In the diagram, w   is the argument of the perihelion, W is the longitude of the ascending node, f is the true anomaly and i is the inclination of the orbit. The critical or key parameter here is h, the angular momentum vector for the orbiting system. It may be useful here to refer to the diagram (b) in the figure used for angular momentum vector (z) in an atomic system:

Getting specific, assuming r and r' are r (radius vector) and d r/dt, respectively, the magnitude h, of the angular momentum vector is:

h = r x r’ =

(y z’ - z y’)

(z x’ - x z’) = (C1 C2 C3)

(x y’ - y z’)

so (r x r’) = (C1/ h, C2/ h, C3/h)

and inserting variables one finds:

C1/ h = sin W sin (i)

C2/ h = - cos W sin (i)

C3/h = cos(i)

Now since the inclination of Earth's orbit to the ecliptic  (i) is known (23.5 deg) and therefore cos(i) can be determined, then sin(i) can be as well.h can be determined, since: h = C3 / cos(i) = (GMm a (1 – e2)12 where all the constants are known (a = semi-major axis of orbit, e = eccentricity of orbit)
(The energy equation is: ½V 2 - u/r = C, and the C's - energy integration constants- are  found from this.)

The student is also able to ascertain that  W  = M - w  (difference between mean anomaly and argument of the perihelion) where M can be obtained from a table based on observations, and w can be obtained using a Fourier expansion of the mean anomaly, M:

e.g. w  = M + (2e – e3 / 4) sin M + 5 e2/4 sin 2M + ... etc.

Once W is known, C1 and C2 and C3 are known, the student will be asked to attempt to compute the position of a planet, say Jupiter, forty or so years in the future. On the basis of this project, the student's final grade may well depend.

Radio Astronomy:

This important area commences with the student introduced to the plasma conditions associated with the motion of charged particles, that give rise to radio wave propagation. For example, he will be expected to recognize and apply the fundamental equation of motion: m (dv/dt) = q(v X B) where q is the charge, and the cross product is for velocity and magnetic induction. The motion is such that the velocity v is always perpendicular to the force acting on the particle, so that:

 dv/dt = q/ m [v X B] is a centripetal acceleration.

Meanwhile, (v⊥ )2 / r = q/ m [v⊥ B]

The quantity r is none other than the gyro-radius. Solving for it one finds:

r = m/ q [v⊥ / B] = v⊥ / (qB/m)

for which one can have either the electron, or ion gyro-frequency. These equations explain the physical basis for the origin of a preponderance of radio waves (i.e. gyro-magnetic emission).  Radiation characteristics will also be covered, and this will include treating the quantity known as the specific intensity i.e.

Il (0,q) =   òo z  Bl(t)  exp [(-tl / cos  q)] dt/ cos  q

where Bl(t)   is the Planck function.    The energy which flows per unit solid angle will then be based upon finding:

dE n =    I  cos  q   d dt  dn 

from which one will wish to obtain the total flux

The essentials of assorted radio telescope properties, especially for antennas, will also be introduced, along with many problems - including practical (i.e. designing a specific antenna to detect an object of given flux, and spectral output etc.).  To this end the student will distinguish between the flux emitted for an isotropic (lossless) antenna, and an anisotropic antenna with the gain (g) subject to the constraint:

ò 4p  g  dw   =   4p 

and the relation between the gain of the antenna and its effective aperture (A) such that:

g( (q ,  φ )  =   4A (q ,  φ)/ l2 

From here, the student will be expected to work out the beam width and beam efficiency of a given antenna, as well as compute the 'brightness temperature' for a localized source, and the antenna temperature (Ta =  1/4p  ò 4p   g T(b)  dw)  where T(b) is the brightness temp

Sensitivity of the antenna will also be considered, as well as other details such as the amplification of high frequency signals. Not the least of these will be the Stokes parameters and problems involving them.  In general the normalized Stokes parameters will always be a combination of  contributions such that:

S [s_i] = S[s] +  S{s’]

Where [s_i] is comprised of 4 components (based on received spectral power):


and s =

[1- d]

and s’ =

[d cos(2Z) cos(2t)]
[d cos(2Z)sin(2t)]
[d sin(2Z)]

where the latter column vector (matrix) discloses the contribution for partial polarization such that:

cos (2Z) = (AR2 – 1)/ (AR2 + 1)

and:  sin(2Z) = 2AR/ (AR2  + 1)

Typical HW problem:

Three radio waves from different objects in space have the following characteristics, where d is the degree of polarization and AR denotes the axial ratio of polarization ellipse. Find the normalized Stokes parameters and the coherency matrix for each:

i) d = 1/2,   AR= 4,  t = 135 deg
ii) d = 1 , AR = -4,  t = p/4
iii) d = 1, AR = -1

Stellar Constitution and Evolution:

As if the Astronomy senior hasn't had enough math already, he will now have to confront the theory of stellar structure.  In this case, the student will be introduced to a variety of differential equations which he'll later be expected to use in the construction of an actual stellar model (which may be 50 percent of his final grade.) He learns that the force of attraction between M(r) e.g. the mass enclosed inside the stellar sphere of radius, r and r dr  (the mass of an element) is the same as that between a mass M(r) at the center and  r dr at r.   By Newton’s law this attractive force is given by:

F = G M(r)  r dr/ r2

Since the attraction due to the material outside r is zero, we should have for equilibrium:

- dP = G M(r) r dr/ r2


Or:     dP/dr = - G M(r) r / r2

 Consider now the mass of the shell between an outer  layer of a given star  and a deeper stellar layer. This is approximately, 4p r2 r dr, provided that dr  (shell thickness) is small. The mass of the layer is the difference between M(r + dr) and M(r) which for a thin shell is:
M(r + dr) -  M(r) = (dM/ dr) dr

Equating the two expressions for the mass of the spherical shell we obtain:

dM/dr = 4p r2 r

The two equations, for dP/dr and dM/dr represent the basic equations of stellar structure, without which the innards of a star would be inaccessible to investigation. A third equation of stellar structure may be derived using by using the equation for dM/dr in combination with the fact that a star’s luminosity is produced through the consumption of its own mass. This may be expressed mathematically as:

dL/dM = e

where e denotes the rate of energy generation. For the proton-proton cycle (for stars like the Sun- and designed for cgs units!): 

e= 2.5 x 106 (r  X2).· (106 /T)2/3 exp[-33.8(106 /T)1/3]

Of course, to construct a stellar model as part of a course project, the student will more likely have to deal with a different 'critter' entirely -  say a star of two solar masses with a convective (as opposed to a radiative) core and with composition: X = 0.65 (i.e. 65% hydrogen), Y = 0.32 (i.e. 32% helium) and Z = 0.03 (i.e. 3 % heavy elements).  Say with an energy generation function:

e= 10-14.2 (r  X X(CN)).· (T20 )

where: X(CN) = 0.01   and :

X =  4.34 x 1025  Z(1 + X)  r  T --3.5

This is none other than the stellar model problem construction I had to complete, and for which I received an 'A' and also an A in the course. The details of the problem were:

Neglect radiation pressure and degeneracy and assume the gas is nearly totally ionized. Utilizing Wrubel's interior integrations and Schwarzschild's and Harm's envelop solutions:

a) Construct a consistent stellar model utilizing the U-V plane fitting technique, and

b) Calculate the following physical properties of the star: L, R, Tc  ,  r c ,  Pc  , and the mass of the convective core,  m c .

Still want to become an Astronomy major?  Just make sure math is in your blood.

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