Essential Radio Astronomy

Appendix B Mathematical Derivations

B.1 Evaluation of Planck’s Sum

Planck’s sum (Equation 2.83) for the average energy per mode of blackbody radiation is


It is convenient to introduce the variable α1/(kT), so


Next consider the quantity


Using the chain rule to take the derivative yields

-ddα[lnn=0exp(-αnhν)] =-[n=0exp(-αnhν)]-1ddα[n=0exp(-αnhν)]





has the form 1+x+x2+=(1-x)-1, so



E =-dln[1-exp(αhν)]-1dα

B.2 Derivation of the Stefan–Boltzmann Law

The Stefan–Boltzmann law for the integrated brightness of blackbody radiation at temperature T (Equation 2.89) is




is Planck’s law and σ is the Stefan–Boltzmann constant. Although the Stefan–Boltzmann law and constant were first determined experimentally, both can be derived mathematically from Planck’s law. For simplicity, define




The quantity


can be expanded in terms of the infinite series

m=0zm= 1+z+z2+z3+
= 1+z(1+z+z2+z3+)
= 1+zm=0zm,
m=0zm= 11-z.



and the integral becomes


Each integral in this series can be integrated by parts three times:

0x3e-mx𝑑x =x3e-mx-m|0-03x2e-mx-m𝑑x=3m0x2e-mx𝑑x,
0x2e-mx𝑑x =x2e-mx-m|0-02xe-mx-m𝑑x=2m0xe-mx𝑑x,
0xe-mx𝑑x =xe-mx-m|0-0xe-mx-m𝑑x=1m0e-mx𝑑x=1m2,

to give




The sum


converges quickly and is the value of the Riemann zeta function ζ(4)=π4/901.082. Thus

0x3dxex-1=π415. (B.1)

Finally, the integrated brightness of blackbody radiation is




is the value of the Stefan–Boltzmann constant.

Similarly, the integral


is needed to evaluate the number density nγ of blackbody photons:


Following the derivation above,





0x2dxex-1=2m=11m3=2(113+123+133+)2.404. (B.2)

B.3 Complex Exponentials

A complex exponential eiϕ, where i2=-1 and ϕ is any dimensionless real variable, is a complex number in which the real and imaginary parts are sines and cosines given by Euler’s formula

eiϕ=cosϕ+isinϕ. (B.3)

Euler’s formula can be derived from the Taylor series

cosϕ =1-ϕ22!+ϕ44!-ϕ66!+,
sinϕ =ϕ-ϕ33!+ϕ55!-ϕ77!+,
eϕ =1+ϕ+ϕ22!+ϕ33!+ϕ44!+.


eiϕ =1+iϕ-ϕ22!-iϕ33!+ϕ44!+iϕ55!-iϕ66!-iϕ77!+

Complex exponentials (or sines and cosines) are widely used to represent periodic functions in physics for the following reasons:

  1. 1.

    They comprise a complete and orthogonal set of periodic functions. This set of functions can be used to approximate any piecewise continuous function, and they are the basis of Fourier transforms (Appendix A.1).

  2. 2.

    They are eigenfunctions of the differential operator—that is, the derivatives of complex exponentials are themselves complex exponentials:


Most physical systems obey linear differential equations, a low-pass filter consisting of a resistor and a capacitor, for example. A sinusoidal input signal will yield a sinusoidal output signal of the same frequency (but not necessarily with the same amplitude and phase), while a square-wave input will not yield a square-wave output. The response to a square-wave input can be calculated by treating the input square wave as a sum of sinusoidal waves, and the filter output is the sum of these filtered sinusoids. This is the reason why periodic waves or oscillations are almost always treated as combinations of complex exponentials (or sines and cosines).

Real periodic signals can be expressed as the real parts of complex exponentials:

cosϕ =Re(eiϕ),
sinϕ =Im(eiϕ).

Adding and subtracting the equations

eiϕ =cosϕ+isinϕ,
e-iϕ =cosϕ-isinϕ

gives the identities

cosϕ=eiϕ+e-iϕ2 (B.4)


sinϕ=eiϕ-e-iϕ2i. (B.5)

The advantage of complex exponentials over the equivalent sums of sines and cosines is that they are easier to manipulate mathematically. For example, you can use complex exponentials to calculate the output spectrum of a square-law detector (Section 3.6.2) without having to remember trigonometric identities. A square-law detector is a nonlinear device whose output voltage is the square of its input voltage. If the input voltage is cos(ωt), the output voltage is

cos2(ωt) =(eiωt+e-iωt2)2

The output spectrum has two frequency components: one at twice the input frequency ω and the other at zero frequency (DC).

B.4 The Fourier Transform of a Gaussian

The normalized Gaussian function is usually written as

f(x)=12πσexp(-x22σ2), (B.6)

where σ is its rms width. To calculate its Fourier transform

F(s)-f(x)exp(-i2πsx)𝑑x, (B.7)

it is easier to use the form f(x)=exp(-πx2), for which σ2=1/(2π). Then

F(s) =-exp(-πx2)exp(-i2πsx)𝑑x (B.8)
=-exp[-π(x2+i2sx+s2-s2)]𝑑x (B.9)
=exp(-πs2)-+is+isexp[-π(x+is)2]d(x+is) (B.10)
=exp(-πs2)-exp(-πx2)𝑑x. (B.11)

To evaluate this one-dimensional integral, break it into the product of two integrals and change one dummy variable from x to y to suggest Cartesian coordinates in a plane:

-exp(-πx2)𝑑x =[-exp(-πx2)dx-exp(-πy2)dy]1/2 (B.12)
=[--exp[-π(x2+y2)]𝑑x𝑑y]1/2. (B.13)

Next transform to polar coordinates r,θ so r2=x2+y2 and dxdy=rdrdθ:

-exp(-πx2)𝑑x=[r=0θ=02πexp(-πr2)r𝑑r𝑑θ]1/2. (B.14)

Finally, substitute uπr2 and du=2πrdr to get

-exp(-πx2)𝑑x=[2πu=0exp(-u)du2π]1/2=[-e-u|0]1/2=1. (B.15)


F(s)=exp(-πs2). (B.16)

The Fourier transform of a Gaussian is a Gaussian.

B.5 The Gaussian Probability Distribution and Noise Voltage

The voltage V of random noise has a Gaussian probability distribution

P(V)=1(2π)1/2σexp(-V22σ2), (B.17)

where P(V)dV is the differential probability that the voltage will be within the infinitesimal range V to V+dV and σ is the root mean square (rms) voltage. The probability of measuring some voltage must be unity, so

-P(V)𝑑V=1. (B.18)

The normalization of P(V) in Equation B.17 can be confirmed by evaluating the integral

-1(2π)1/2σexp(-V22σ2)𝑑V =201(2π)1/2σexp(-V22σ2)𝑑V (B.19)
=[2(2π)1/2σ]0exp(-V22σ2)𝑑V. (B.20)

Equation B.15 immediately yields the definite integral

0exp(-a2x2)𝑑x=π1/22a. (B.21)

Substituting a2=(2σ2)-1 gives the desired result:

-P(V)𝑑V=[2(2π)1/2σ](π1/22)(2σ2)1/2=1. (B.22)

The rms (root mean square) Σ of a normalized distribution is defined by

Σ2V2-V2. (B.23)

For the symmetric Gaussian distribution, V=0, so

Σ2 =V2=-V2P(V)𝑑V (B.24)
=20V21(2π)1/2σexp(-V22σ2)𝑑V (B.25)
=[2(2π)1/2σ]0V2exp(-V22σ2)𝑑V. (B.26)

The definite integral

0x2exp(-a2x2)𝑑x=π1/24a3 (B.27)

can be derived by integrating Equation B.21 by parts. Inserting a2=(2σ2)-1 yields

Σ2=V2=[2(2π)1/2σ](π1/24)(2σ2)3/2=σ2, (B.28)

confirming that σ in Equation B.17 is the rms of the Gaussian distribution.

B.6 The Probability Distribution of Noise Power

A square-law detector multiplies the input voltage V by itself to yield an output voltage Vo=V2 that is proportional to the input power. The input voltage distribution is a Gaussian with rms σ (Equation B.17),

P(V)=1(2π)1/2σexp(-V22σ2). (B.29)

The same value of Vo=V2 is produced by both positive and negative values of V and P(V)=-P(V), so

Po(Vo)dVo=2P(V)dV (B.30)

for all Vo0. Because dVo=2VdV,

Po(Vo)=[Vo-1/2(2π)1/2σ]exp(-Vo2σ2) (B.31)

for Vo0. The distribution of detector output voltage is sharply peaked near Vo=0 and has a long exponentially decaying tail (Figure 3.33).

The mean detector output voltage follows from Equation B.28: Vo=V2=σ2.

The rms σo of the detector output voltage is

σo2=Vo2-Vo2, (B.32)


Vo2 =0Vo2Po(Vo)𝑑Vo=0V42P(V)𝑑V (B.33)
=[2(2π)1/2σ]0V4exp(-V22σ2)𝑑V. (B.34)

Integrating Equation B.27 by parts yields the definite integral

0x4exp(-a2x2)𝑑x=3π1/28a5, (B.35)

and substituting a2=(2σ2)-1 gives

Vo2=[2(2π)1/2σ](3π1/28)(2σ2)5/2=3σ4. (B.36)


σo2=Vo2-Vo2=3σ4-(σ2)2=2σ4. (B.37)

The rms σo=21/2σ2 of the detector output voltage is 21/2 times the mean output voltage σ2. The rms uncertainty in each independent sample of the measured noise power is 21/2 times the mean noise power. If N1 independent samples are averaged, the fractional rms uncertainty of the averaged power is (2/N)1/2. This result is the heart of the ideal radiometer equation (Equation 3.154). According to the central limit theorem, the distribution of these averages approaches a Gaussian as N becomes large.

B.7 Evaluation of the Free–Free Pulse Energy Integral

The integral in Equation 4.22 is

0π/2cos4ψdψ =0π/2cos2ψ(1-sin2ψ)𝑑ψ
=0π/2cos2ψdψ-0π/2cos2ψsin2ψdψ. (B.38)

We have already found that cos2ψ=1/2 so 0π/2cos2ψdψ=π/4. Integrate the remaining integral by parts using

ucos2ψsinψ  and  vdvsinψdψ.


du=cos3ψ-2sin2ψcosψ  and  v=-cosψ,


0π/2cos2ψsin2ψdψ =-cos2ψsinψcosψ|0π/2
which has the same integral on both sides, so
0π/2cos2ψsin2ψdψ =130π/2cos4ψdψ.

Using Equation B.38 we get

0π/2cos4ψdψ =π4-130π/2cos4ψdψ,
430π/2cos4ψdψ =π4,


0π/2cos4ψdψ=3π16. (B.39)

B.8 The Nonrelativistic Maxwellian Speed Distribution

Let v|v| be the speed of a particle (e.g., an electron) of mass m in a gas in LTE at temperature T. From thermodynamics, recall that the average kinetic energy is kT/2 per degree of freedom (e.g., per spatial coordinate for a single particle), so

mvx22=mvy22=mvz22=kT2, (B.40)
v2=vx2+vy2+vz2=3kTm. (B.41)

Collisions eventually bring the gas into LTE, leading to identical Gaussian distributions (Appendix B.5) for vx, vy, and vz. Writing out only the x-coordinate distribution P(vx) yields

P(vx)=12πσxexp(-vx22σx2), (B.42)

where σx is the rms (root mean square) value of vx. The definition of this rms is

σx2vx2 =-vx2P(vx)𝑑vx=-vx22πσxexp(-vx22σx2)𝑑vx (B.43)
=12πσx12π(12σx2)-3/2=kTm, (B.44)


P(vx)=12π(mkT)1/2exp(-mvx22kT). (B.45)

In three dimensions, by isotropy,

P(vx,vy,vz)dvxdvydvz=P(vx)P(vy)P(vz)dvxdvydvz, (B.46)
P(vx,vy,vz)=(m2πkT)3/2exp(-mv22kT). (B.47)

All velocities in the spherical shell of radius v=(vx2+vy2+vz2)1/2 correspond to the speed v, so

f(v)=4πv2P(vx,vy,vz), (B.48)
f(v)=4v2π(m2kT)3/2exp(-mv22kT). (B.49)

This is the nonrelativistic Maxwellian distribution f of speeds vc for particles of mass m at temperature T. If we normalize the speeds by the rms speed (3kT/m)1/2, the Maxwellian speed distribution looks like Figure 4.6.