Introduction to Millimeter/Sub-Millimeter Astronomy

106 Pages·2009·1.469 MB·English

by  Wilson, T. L

Introduction to Millimeter/Sub-Millimeter
Astronomy
T. L. Wilson1,2
9 1 European Southern Observatory,K-Schwarzschild-Str.2, D85748 Garching,
0 [email protected]
0 2 Max-Planck-Institut-f-Radioastronomie
2
r 1 Introduction
a
M
This chapter provides an introduction to the following lectures. It is based
on nine lectures given at the Saas Fee school in March 2008. Much, but not
3
all of this material is contained in [38] and [53], but with more more em-

] phasis on current research in millimeter/sub-millimeter astronomy. In the
A
following,the basics ofradiativetransfer,receivers,antennas,interferometry
G radiationmechanismsandmoleculesarepresented.Thefieldofmm/sub-mm
. astronomy has become very large, so this presentation contains only a few
h
derivations. The astute reader will note that Maxwell’s Equations do not
p
- appear at all, although the results are based on these. In some cases, the
o approach is to quote a result followed by an example. This contribution is
r
t aimed at graduate students with a good background in physics and astron-
s
omy.Somecommontermsareusedbut′′jargon′′hasbeenavoidedasmuchas
a
[ possible. References are given, for the most part, to more recent work where

citations to earlier work can be found. The units are mostly CGS with some
1
v SI units. This follows the usage in the astronomy literature. One topic not
2 covered here is polarization; see [38] and [46] for an introduction.
6 The field of mm/sub-mm astronomy began only in the 1960’s, but the
5 richness of the results has justified this series of lectures. Millimeter/sub-
0
mm measurements require excellent weather, very accurate antennas, and
.
3 sensitive receivers. The interpretation of these data requires a knowledge of
0
atomic and molecular physics, radiative transfer and interstellar chemistry.
9
By number, about 90% of molecular clouds consist of H . However, the de-
0 2
: termination of local densities and column densities of molecular hydrogen,
v
H must be indirect since H does not emit spectral lines in cooler clouds.
i 2 2
X The Schr¨odinger equation governs all chemistry, but the conditions in the
r interstellarmedium arevery different(and muchmore varied)than those on
a
earth. Earth-bound chemistry is only a subset of the more general interstel-
larchemistry;itisdominatedbynon-equilibriumprocessesthatoccuratlow
temperatures and densities, so determinations of collision and reaction rates
are needed. A project devoted to this goal is the ′′Molecular Universe′′ (see
the website fordetails). Foranoverviewofinterstellarchemistry,the reader
is referred to the short presentation [21] or the monograph [47]. Although
interstellar chemistry plays a very important role in molecular line astron-
2 T. L. Wilson
omy, it is not included due to space limitations. Another glaring omission is
a treatment of the Cosmic Microwave Background (CMB), since this is not
treated in the following two lectures. Technically, the CMB emission is not
weak, but does fill the entire sky, so special techniques must be employed.
Mm/sub-mm astronomy is one of the most important tools to study the
birth of stars [45], [37] and star formation in galaxies [44]. Stars form in
molecular clouds where extinction is large. Thus, near infrared and optical
studies are of limited value. At high red shifts, many active star forming
galaxies and Active Galactic Nuclei (AGN’s) are enshrouded by dust [43].
Synchrotron emission at cm wavelengths from AGN’s allows sub-arcsecond
resolution images of relativistic electrons moving in B fields [42]. However,
the imaging ofdust and molecules oncomparablescales is just beginning, so
this is an area where important contributions can be made.
The closest example of a supermassive Black Hole is Sgr A∗, 8.5 kpc (1
kpc=3.08 1021 cm) from the Sun. Sgr A∗ is quiescent at present, but is
×
thoughttobesimilartothe′′engines′′ thatpowerAGN’s.Theradioemission
from Sgr A∗ is interpreted as optically thick synchrotron emission, so with
mm/sub-mm Very long Baseline interferometry, gas on Astronomical Unit
(1 AU=1.5 1013 cm) scales, close to the Schwarzschild radius, can be
×
sampled.
Inthefieldofstarformation,webelievethatallofthephysicalprinciples
areknown.However,starformationiscomplexandmeasurementsareneeded
to determine which processes are dominant and which can be neglected. In
contrast, for studies of the early universe, Black Holes and AGN’s not all of
the relevant physical laws may be known. As in all of astronomy, measure-
ments often lead to unexpected results that may have far reaching effects on
our understanding of fundamental physical laws.
Thedominantcontinuumradiationmechanisminthemm/sub-mmwave-
length range is dust emission. This differs from free-free and synchrotron
emission, which is commonly encountered at centimeter wavelengths [13].
Spectralline radiationisdominatedbythermalandquasi-thermalmolecular
radiation,althoughthere area few importantatomiclines ofcarbon,oxygen
and nitrogen.
The sensitivity atmm/sub-mm wavelengthshas become about100 times
better than was the case in the 1960’s [20]. However sensitivity alone is not
enough to transform the field. Rather, high resolution images are needed.
This will change when the Atacama Large Millimeter Array (ALMA) begins
operation.ALMAwill initially operatebetween3.5mmand0.4mm.ALMA
has a unique combination of high angular resolution and high sensitivity.
This will allow ALMA to transform the field of mm/sub-mm astronomy. A
summary of the science planned for ALMA is to be found in [5].
Millimeter/Sub-Millimeter Astronomy 3
2 Some Background
In this section, we review the basics needed in following sections (see [38]).
Electromagnetic radiation in the radio window can be interpreted as a wave
phenomenon, i. e. in term of classical physics. When the scale of the system
involved is much larger than a wavelength, we can consider the radiation to
travelin straightlines or rays. The power,dP, interceptedby a infinitesimal
surface dσ is
dP =I cosθdΩdσdν (1)
ν
where
dP =power, in watts,
dσ =area of surface, m2,
dν =bandwidth, in Hz,
θ =angle between the normal to dσ and the direction to dΩ,
I =brightness or specific intensity, in Wm−2Hz−1sr−1.
ν
Equation (1) is the definition of the brightness I . Quite often the term
ν
intensity or specific intensity I is used instead of the term brightness. We
ν
will use all three designations interchangeably.
The total flux of a source is obtained by integrating (1) over the total
solid angle Ω subtended by the source
s
S = I (θ,ϕ)cosθdΩ, (2)
ν ν
Z
Ωs
andthisfluxdensityismeasuredinunitsofWm−2Hz−1.Sincethefluxden-
sity of astronomical sources is usually very small, a special unit, the Jansky
(hereafter Jy) has been introduced
1Jy=10−26Wm−2Hz−1 =10−23ergs−1cm−2Hz−1. (3)
As long as the surface element dσ covers the ray bundle completely, the
power remains constant:
dP =dP . (4)
1 2
From this, we can easily obtain
I =I (5)
ν1 ν2
sothatthebrightnessisindependentofthedistance.ThetotalfluxS density
ν
shows the expected dependence of 1/r2.
Another useful quantity related to the brightness is the radiation energy
density u in units of erg cm−3. From dimensional analysis, u is intensity
ν ν
divided by speed. Since radiation propagates with the velocity of light c, we
have for the spectral energy density per solid angle
4 T. L. Wilson
1
u (Ω)= I . (6)
ν ν
c
If integrated over the whole sphere, 4π steradian, (Eq. 6) will yeild the total
spectral energy density
1
u = u (Ω)dΩ = I dΩ. (7)
ν ν ν
c
Z Z
(4π) (4π)
2.1 Radiative Transfer
The equation of transfer is
dI
ν
= κ I +ε . (8)
ν ν ν
ds −
Thelinearabsorptioncoefficientκ andtheemissivityε areindependent
ν ν
of the intensity I leading to the above form for dI .
ν ν
In Thermodynamic Equilibrium (TE) there is complete equilibrium of
the radiation with its surroundings, the brightness distribution is described
by the Planck function, which depends only on the temperature T, of the
surroundings. The properties of the Planck function will be described in the
next section.
In Local Thermodynamic Equilibrium (LTE) Kirchhoff’s law holds
ε
ν
=B (T) (9)
ν
κ
ν
This is independent of the material, as is the case with complete thermody-
namic equilibrium. In general however, I will differ from B (T).
ν ν
If we define the optical depth dτ by
ν
dτ = κ ds (10)
ν ν
−
or
s
τ (s)= κ (s)ds, (11)
ν ν
Z
s0
then the equation of transfer (8) can be written as
1 dI dI
ν ν
= =I B (T) . (12)
ν ν
−κ ds dτ −
ν ν
The solution of (12) is
Millimeter/Sub-Millimeter Astronomy 5
τν(s)
I (s)=I (0)e−τν(s)+ B (T(τ))e−τdτ . (13)
ν ν ν
Z
0
If the medium is isothermal,
T(τ)=T(s)=T =const.
the integral is
I (s)=I (0)e−τν(s)+B (T)(1 e−τν(s)) . (14)
ν ν ν
−
For a largeopticaldepth, thatis forτ (0) , (14) in LTEapproachesthe
ν
→∞
limit
I =B (T). (15)
ν ν
This is case for planets and the 2.7 K microwavebackground.The difference
between I (s) and I (0) gives
ν ν
∆I (s)=I (s) I (0)=(B (T) I (0))(1 e−τ). (16)
ν ν ν ν ν
− − −
Eq. 16 represents an on-source minus an off-source measurement.
2.2 Black Body Radiation
and Brightness Temperature
The spectraldistribution of the radiationof a black body in thermodynamic
equilibrium is given by the Planck law
2hν3 1
B (T)= .
ν c2 ehν/kT 1
−
If hν kT, one obtains Rayleigh-Jeans Law.
≪
2ν2
B (ν,T)= kT . (17)
RJ c2
This is an expression of classical physics in which there is an energy kT per
mode.Theterm 2ν2 isthedensityofstatesfor3dimensions.Inthemillimeter
c2
andsubmillimeterrange,onefrequentlydefinesaradiationtemperature,J(T)
as
c2 hν 1
J(T)= I = . (18)
2kν2 k ehν/kT 1
−
Inserting numerical values for k and h, we find that the Rayleigh-Jeans
relation holds for frequencies
ν T
20.84 . (19)
GHz ≪ K
(cid:18) (cid:19)
6 T. L. Wilson
It can thus be used for all thermal radio sources except perhaps for low
temperatures in the mm/sub-mm range.
In the Rayleigh-Jeans relation, the brightness and the thermodynamic
temperature of the black body that emits this radiation are strictly propor-
tional (17). This feature is so useful that it has become the custom in radio
astronomy to measure the brightness of an extended source by its bright-
ness temperatureT .Thisisthe temperaturewhichwouldresultinthe given
b
brightness if inserted into the Rayleigh-Jeans law
c2 1 λ2
T = I = I . (20)
b 2kν2 ν 2k ν
Combining (2) with (20), we have
2kν2
S = T D Ω . (21)
ν c2 b
For a Gaussian source, this relation is
2 −2
S θ λ
ν
=0.0736T (22)
b
Jy arcseconds mm
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
That is, if the flux density S and the source size are known, then the true
ν
brightnesstemperature,T ,of the sourcecanbe determined.The conceptof
b
temperature in radio astronomy has given rise to confusion. If one measures
S and the apparent source size, Eq. 22 allows one to calculate the main
ν
beam brightness temperature, T . Details of this procedure will be given
MB
in Section 5. The performance of coherent receivers is characterized receiver
noise temperature; see Section 3.2.1 for details. The combination of receiver
andatmosphereischaracterizedbysystemnoisetemperature;thisisdiscussed
in Section 6.1 and following.
IfI isemittedbyablackbodyandhν kT then(20)givesthethermo-
ν
≪
dynamictemperatureofthe source,avalue thatis independentofν.Ifother
processesareresponsibleforthe emissionofthe radiation,T willdepend on
b
the frequency; it is, however,still a useful quantity and is commonly used in
practical work.
This is the caseeven if the frequency is so high that condition(19) is not
valid. Then (20) can still be applied, but T is different from the thermody-
b
namic temperature of a black body. However, it is rather simple to obtain
the appropriate correction factors.
It is also convenient to introduce the concept of brightness temperature
into the radiative transfer equation (16). Formally one can obtain
c2
J(T)= (B (T) I (0))(1 e−τν(s)).
2kν2 ν − ν −
UsuallycalibrationproceduresallowonetoexpressJ(T)asT.Thismeasured
quantity is referred to as T∗, the radiation temperature, or the brightness
R
Millimeter/Sub-Millimeter Astronomy 7
temperature,T . Inthe cm wavelengthrange,one canapply (20) to (12) and
b
one obtains
dT (s)
b
=T (s) T(s) , (23)
b
dτ −
ν
where T(s) is the thermodynamic temperature of the medium at the posi-
tion s. The general solution is
τν(s)
T (s)=T (0)e−τν(s)+ T(s)e−τdτ . (24)
b b
Z
0
If the medium is isothermal, this becomes
T (s)=T (0)e−τν(s)+T (1 e−τν(s)) . (25)
b b
−
2.3 The Nyquist Theorem and the Noise Temperature
We now relate voltage and temperature; this is essential for the analysis of
receiver systems limited by noise. The average power per unit bandwidth
produced by a resistor R is
v2 1
P = iv = h i = v2 , (26)
ν h i 2R 4Rh Ni
wherev(t)isthe voltagethatisproducedbyiacrossR,and indicatesa
h···i
time average. The first factor 1 arises from the condition for the transfer of
2
maximumpowerfromRoverabroadrangeoffrequencies.Thesecondfactor
1 arisesfromthe time averageofv2.Ananalysisofthe randomwalkprocess
2
shows that
v2 =4RkT . (27)
h Ni
Inserting this into (26) we obtain
P =kT . (28)
ν
Eq.(28) can also be obtained by a formulation of the Planck law for one
dimension and the Rayleigh-Jeans limit. Then, the available noise power of
a resistor is proportional to its temperature, the noise temperature T , and
N
independent of the value of R. Throughout the whole radio range, from the
longest waves to the far infrared region the noise spectrum is white, that is,
its power is independent of frequency. Since the impedance of a noise source
should be matched to that of the receiver, such a noise source can only be
matched over some finite bandwidth.
8 T. L. Wilson
2.3.1 Hertz Dipole and Larmor Formula
In contrastto thermal noise, radiation from an antenna has a definite polar-
izationanddirection.Oneexample(in electromagnetismthere arenosimple
examples!) is a Hertz dipole. The total power radiated from a Hertz dipole
carrying an oscillating current I at a wavelength λ is
2c I∆l 2
P = . (29)
3 2λ
(cid:18) (cid:19)
For the Hertz dipole, the radiation is linearly polarized with the E field
along the direction of the dipole. The radiation pattern has a donut shape,
with a cylindrically symmetric maximum perpendicular to the axis of the
dipole. Along the direction of the dipole, the radiation field is zero. One
can use collections of dipoles, driven in phase, to restrict the direction of
radiation. As a rule of thumb, Hertz dipole radiators have the best radiative
efficiency when the wavelength of the radiation is roughly the size of the
dipole. Following Planck, one can produce the the Black Body law from a
collection of (quantized) Hertz dipole oscillators with random phases.
There are similarities between Hertz dipole radiation and radiation from
atoms. Following Larmor,the power radiated by a single electron oscillating
with a velocity v(t) is
2e2v˙(t)2
P(t)= . (30)
3 c3
Expressing v˙ = x¨, we obtain an average power, emitted over one period of
sinusoidal oscillation, x=sin2πνt.
64π4 ex 2
P = ν4 0 . (31)
h i 3c3 mn 2
(cid:16) (cid:17)
Using ex0 = µ one arrives at the expressionfor spontaneous emission from
2 | |
a quantum mechanicalsystem. This is presentedagainin Eq.132.It is often
the case that quantum mechanical expressions and classical correspond to
within factors of a few.
3 Signal Processing
and Stationary Stochastic Processes
Next, we presentthe basics of signalprocessingand noise analysis needed to
understand the properties of radiometers [38]. Fourier methods are of great
value in analyzing receiver properties [7]. A review of submillimeter receiver
systems is to be found in [39].
TheconceptofspectralpowerdensitywasintroducedinEq.26inapracti-
calexample.Radioreceiversaredevicesthatmeasurespectralpowerdensity.
Millimeter/Sub-Millimeter Astronomy 9
Sincethesignalsaredominatedbynoise,statisticalanalysesareneeded.The
mostimportantof these statisticalquantities is the probabilitydensity func-
tion, p(x), which gives the probability that at any arbitrarymoment of time
the value of the process x(t) falls within an interval (x 1dx,x+ 1dx). For
− 2 2
a stationary random process, p(x) will be independent of the time t.
The expected value E x or mean value of the randomvariablex is given
{ }
by the integral
∞
E x = xp(x)dx (32)
{ }
Z
−∞
and, by analogy, the expectation value E f(x) of a function f(x) is given
{ }
by
∞
E f(x) = f(x)p(x)dx . (33)
{ }
Z
−∞
A digression: the trend in all forms of communications (including radio
astronomy!)istowarddigitalprocessing.Ingeneral,adigitizedfunctionmust
be sampled at regular intervals. Assume that the input signal extends from
zero Hz to ν Hz (this is referred to the video band). Then the bandwidth,
0
∆ν ,isν andmaximumfrequencyisν .Ifwepicturetheinputasacollection
0 0 0
of sine waves, it is clear that the sampling rate must be at least ν = 2∆ν
0
to characterize the sinusoid with the highest frequency, sin2πν t. Thus the
0
sample rate mustbe twice the bandwidth ofthe videoinput. This is referred
to as the Nyquist Sampling Rate. This is a minimum; a higher sampling
rate can only improve the characterization of the input. A higher sampling
rate (′′oversampling′′) will allow the input to be better characterized, thus
giving a better Signal-to-Noise ratio (S/N) ratio. The sampling functions
mustoccupyanextremelysmalltime intervalcomparedtothe time between
samples.Ifonlyaportionoftheinputfunctionisretainedinthequantization
and sampling process, information is lost. This results in a lowering of the
S/N ratio.
Atpresent,commerciallyavailabledigitizerscansamplea 1.5GHz band-
width with 8 bit quantization. This allows digital systems to reach within a
few percent of the sensitivity of analog systems, but with much greater sta-
bility. After digitization, one can maintain that all of the following processes
are ′′just arithmetic′′. The quality of the data (i. e. the S/N) will depend on
the analog receiver elements. The digital parts of a receiver can lower S/N
ratios, but not raise them!
10 T. L. Wilson
3.1 Square Law Detectors
In radio receivers, noise is passed through a device that produces an output
signal y(t) which is proportionalto the power in a given input v(t):
y(t)=av2(t). (34)
This involves an evaluation of the integral
E y(t) = E av2(t) = a +∞v2e−v2/2σ2dv
{ } { } σ√2π
Z−∞
There are standardapproachs used to evaluate this expression.The result is
E y(t) =E v2(t) =aσ2 (35)
{ } { } v
For the evaluation of E y2(t) , one must calculate
{ }
E y(t)2 =E v4(t) = 1 +∞v4e−v4/2σ2dv
{ } { } σ√2π
Z−∞
The result of this integration is
E y2(t) =3a2σ4 (36)
{ } v
and hence
σ2 =E y2(t) E2 y(t) =2E2 y(t) . (37)
y { }− { } { }
3.2 Limiting Receiver Sensitivity
A receiver must be sensitive, that is, be able to detect faint signals in the
presenceofnoise.Therearelimitsforthissensitivity,sincethereceiverinput
and the receiver itself are affected by noise. Even when no input source is
connectedtoareceiver,thereisanoutputsignal,sinceanyreceivergenerates
thermal noise. This noise is amplified together with the signal. Since signal
andnoisehavethesamestatisticalproperties,thesecannotbedistinguished.
To analyze the performance of a receiver we will use the model of an ideal
receiver producing no internal noise, but connected simultaneously to two
noise sources,one for the external source noise and a second for the receiver
noise.To be useful, receiversmust increasethe input power level.The power
per unit bandwidth, P , entering a receiver can be characterized by a tem-
ν
perature, as given by Eq.28, P = kT. Furthermore, it is always the case
ν
that the noise contributions from source, atmosphere, ground and receiver,
T , are additive,
i
T = T
sys i
X
An often-used figure of merit is the Noise Factor, F. This is defined as