Essential Radio Astronomy

Appendix E Essential Equations

The specific intensity Iν of radiation is defined by

IνdP(cosθdσ)dνdΩ, (\ref{eqn:SIorSB})

where dP is the power received by a detector with projected area (cosθdσ) in the solid angle dΩ and in the frequency range ν to ν+dν. Likewise Iλ is the brightness per unit wavelength:

IλdP(cosθdσ)dλdΩ. (\ref{eqn:ilambdadef})

These two quantities are related by

IλIν=|dνdλ|=cλ2=ν2c. (\ref{eqn:ilambdaoverinu})

The flux density Sν of a source is the spectral power received per unit detector area:

SνsourceIν(θ,ϕ)cosθdΩ. (\ref{eqn:fluxdensity})

If the source is compact enough that cosθ1 then

SνsourceIν(θ,ϕ)dΩ. (\ref{eqn:simplefluxdensity})

The MKS units of flux density are Wm-2Hz-1; 1jansky(Jy)10-26Wm-2Hz-1.

The spectral luminosity Lν of a source is the total power per unit frequency radiated at frequency ν; its MKS units are W Hz-1. In free space and at distances d much greater than the source size, the inverse-square law

Lν=4πd2Sν (\ref{eqn:speclum})

relates the spectral luminosity of an isotropic source to its flux density.

The linear absorption coefficient at frequency ν of an absorber is defined as the probability dP(ν) that a photon will be absorbed in a layer of thickness ds:

κ(ν)dP(ν)ds. (\ref{eqn:kappanu})

The opacity or optical depth τ is defined as the sum of those infinitesimal probabilities through the absorber, starting at the source end:

τsoutsin-κ(s)ds. (\ref{eqn:opacity})

The emission coefficient at frequency ν is the infinitesimal increase dIν in specific intensity per infinitesimal distance ds:

jνdIνds. (\ref{eqn:EmissCoeff})

The equation of radiative transfer is

dIνds=-κIν+jν. (\ref{eqn:RadXferEq})

For any substance in Local Thermodynamic Equilibrium (LTE), Kirchhoff’s law connects the emission and absorption coefficients via the specific intensity Bν of blackbody radiation:

jνκ=Bν(T). (\ref{eqn:KirchhoffsLaw})

The brightness temperature of a source with any specific intensity Iν is defined as

Tb(ν)Iνc22kν2. (\ref{eqn:BrightnessTemp})

For an opaque body in LTE, Kirchhoff’s law connects the emission coefficient eν (the spectral power per unit area emitted by the body divided by the spectral power per unit area emitted by a blackbody) to the absorption coefficient aν (fraction of radiation absorbed by the body) and the reflection coefficient rν (fraction of radiation reflected by the body):

eν=aν=1-rν. (\ref{eqn:KirchhoffsLaw2})

The spectral energy density of radiation is

uν=1cIνdΩ. (\ref{eqn:RadEdensity})

The Rayleigh–Jeans approximation for the specific intensity of blackbody radiation when hνkT is

Bν=2kTν2c2=2kTλ2. (\ref{eqn:RJLaw})

The energy of a photon is

E=hν. (\ref{eqn:photonenergy})

Planck’s equation for the specific intensity of blackbody radiation at any frequency is

Bν=2hν3c21exp(hνkT)-1. (\ref{eqn:PlanckLaw})

The total intensity of blackbody radiation is

B(T)0Bν(T)dν=σT4π, (\ref{eqn:IntBright})

where the Stefan–Boltzmann constant σ is defined by

σ2π5k415c2h35.67×10-5ergcm2sK4sr. (\ref{eqn:SBConstant})

The total energy density of blackbody radiation is

u=4σT4c=aT4, (\ref{eqn:BBEdensity})

where a4σ/c7.56577×10-15ergcm-3K-4 is the radiation constant.

The photon number density of blackbody radiation is

(nγcm-3)20.3(TK)3. (\ref{eqn:bbphotondensity})

The mean photon energy of blackbody radiation is

Eγ 2.70kT. (\ref{eqn:meanphotonenergy})

The frequency of the peak blackbody brightness per unit frequency Bν is

(νmaxGHz)59(TK). (\ref{eqn:bbnupeak})

The wavelength of the peak blackbody brightness per unit wavelength Bλ is given by Wien’s displacement law:

(λmaxcm)0.29(TK)-1. (\ref{eqn:WienLaw})

The flux density of isotropic radiation is

Sν=πIν. (\ref{eqn:isotfluxdensity})

The Nyquist approximation for the spectral power generated by a warm resistor in the limit hνkT is

Pν=kT. (\ref{eqn:NyquistLaw})

At any frequency, the exact Nyquist formula is

Pν=hνexp(hνkT)-1. (\ref{eqn:QuantumNyquistLaw})

The critical density needed to close the universe is

ρc=3H028πG8.6×10-30gcm-3. (\ref{eqn:closuredensity})

Redshift z is defined by

zλo-λeλe=λoλe-1=νeνo-1, (\ref{eqn:redshift})

where λe and νe are the wavelength and frequency emitted by a source at redshift z, and λo and νo are the observed wavelength and frequency at z=0.

Redshift z and expansion scale factor a are related by

(1+z)=a-1. (\ref{eqn:scalesize})

The CMB temperature at redshift z is

T=T0(1+z). (\ref{eqn:tcmb})

The radiated electric field at distance r from a charge q at angle θ from the acceleration v˙ is

E=qv˙sinθrc2. (\ref{eqn:LarmorEField})

In a vacuum, the Poynting flux, or power per unit area, is

|S|=c4πE2. (\ref{eqn:PoyntingFlux})

The total power emitted by an accelerated charge is given by Larmor’s formula

P=23q2v˙2c3, (\ref{eqn:LarmorPower})

which is valid only if vc.

Exponential notation for trigonometric functions is

e-iωt=cos(ωt)-isin(ωt). (\ref{eqn:expnotation})

Electric current is defined as the time derivative of electric charge:

Idqdt. (\ref{eqn:currentdef})

The power pattern of a short dipole antenna is

Psin2θ. (\ref{eqn:DipolePattern})

The power emitted by a short (lλ) dipole driven by a current I=I0e-iωt is

P=π23c(I0lλ)2. (\ref{eqn:DipolePower})

Radiation resistance is defined by

R2PI02. (\ref{eqn:RadResist})

Energy conservation implies the average power gain of any lossless antenna is

G=1 (\ref{eqn:AvgGain})


sphereGdΩ=4π. (\ref{eqn:gainintegral})

The beam solid angle is defined by

ΩA4πGmax=1Gmax4πG(θ,ϕ)dΩ. (\ref{eqn:txBSA})

The effective area of an antenna is defined by

Ae2Pν/Sν, (\ref{eqn:EffectiveArea})

where Pν is the output power density produced by an unpolarized point source of total flux density Sν.

The average effective area of any lossless antenna is

Ae=λ24π. (\ref{eqn:AvgArea})

Reciprocity implies

G(θ,ϕ)Ae(θ,ϕ). (\ref{eqn:Reciprocity})

Reciprocity and energy conservation imply

Ae(θ,ϕ)=λ2G(θ,ϕ)4π. (\ref{eqn:GainArea})

Antenna temperature is defined by

TAPνk. (\ref{eqn:AntennaTemp})

The antenna temperature produced by an unpolarized point source of flux density S is

TA=AeS2k. (\ref{eqn:AntSens})

If Ae2761m2, the point-source sensitivity is 1KJy-1.

For a uniform compact source of brightness temperature Tb covering solid angle Ωs,

TATb=ΩsΩA. (\ref{eqn:TaCompact})

The main beam solid angle is defined by the integral over the main beam to the first zero only:

ΩMB1GmaxMBG(θ,ϕ)𝑑Ω (\ref{eqn:MainBeamSolidAngle})

and is used in the definition of main beam efficiency:

ηBΩMBΩA. (\ref{eqn:MainBeamEfficiency})

The height z at axial distance r above the vertex of a paraboloidal reflector of focal length f is

z=r24f. (\ref{eqn:parabola})

The far-field distance of an aperture of diameter D used at wavelength λ is

Rff2D2λ. (\ref{eqn:farfield})

In the far field, the electric field pattern of an aperture antenna is the Fourier transform of the aperture illumination:

lsinθ, (\ref{eqn:ldeff})
uxλ, (\ref{eqn:defineU})
f(l)=apertureg(u)e-i2πlu𝑑u. (\ref{eqn:fieldpattern})

The power pattern of a uniformly illuminated linear aperture is

P(θ)sinc2(θDλ), (\ref{eqn:uniformpowerpattern})

where sinc(x)sin(πx)/(πx), and the half-power beamwidth is

θHPBW0.89λD. (\ref{eqn:beamwidth})

The half-power beamwidth (HPBW) of a a typical radio telescope with tapered illumination is

θHPBW1.2λD. (\ref{eqn:HPBW})

The two-dimensional aperture field pattern is

f(l,m)--g(u,v)e-i2π(lu+mv)dudv, (\ref{eqn:2dfieldpattern})

where m is the y-axis analog of l on the x-axis, and vy/λ. The electric field pattern of a two-dimensional aperture is the two-dimensional Fourier transform of the aperture field illumination.

The power pattern of a uniformly illuminated rectangular aperture is

G4πDxDyλ2sinc2(θxDxλ)sinc2(θyDyλ). (\ref{eqn:2DPowerPattern})

Aperture efficiency is defined by

ηAmax(Ae)Ageom. (\ref{eqn:ApertureEfficiency})

The beam solid angle of a Gaussian beam is

ΩA=(π4ln2)θHPBW21.133θHPBW2. (\ref{eqn:gaussbsa})

The surface efficiency ηs of a reflector whose surface errors ϵ have rms σ is given by the Ruze equation:

ηs=exp[-(4πσλ)2]. (\ref{eqn:SurfaceEfficiency})

Noise temperature is defined by

TNPνk. (\ref{eqn:NoiseTemp})

The system noise temperature is the sum of noise contributions from all sources:

Ts=Tcmb+Trsb+ΔTsource+[1-exp(-τA)]Tatm+Tspill+Tr+. (\ref{eqn:SystemNoise})

The ideal total-power radiometer equation is

σTTs[1Δντ]1/2. (\ref{eqn:IdealRadiometer})

The practical total-power radiometer equation includes the effects of gain fluctuations:

σTTs[1Δντ+(ΔGG)2]1/2. (\ref{eqn:Radiometer})

The Dicke-switching radiometer equation is

σT2Ts[1Δντ]1/2. (\ref{eqn:DickeRadiometer})

The rms confusion caused by unresolved continuum sources in a Gaussian beam with HPBW θ at frequency ν is

(σcmJybeam-1){0.2(νGHz)-0.7(θarcmin)2(θ>0.17arcmin),2.2(νGHz)-0.7(θarcmin)10/3(θ<0.17arcmin). (\ref{eqn:rmsconfusion})

Individual sources fainter than the confusion limit 5σc cannot be detected reliably.

Radiometer input noise temperature Tr can be measured by the Y-factor method; it is

Tr=Th-YTcY-1. (\ref{eqn:YfactorT})

The response of a two-element interferometer to a source of brightness distribution Iν(s^) is the complex visibility

𝒱ν=Iν(s^)exp(-i2πbs^/λ)dΩ. (\ref{eqn:Complexvis})

To minimize bandwidth smearing in bandwidth Δν, the image angular radius Δθ should satisfy

ΔθΔνθsν. (\ref{eqn:bandsmear})

To minimize time smearing in an image of angular radius Δθ the averaging time should satisfy

ΔθΔtθsP2πθs1.37×104s. (\ref{eqn:timesmear})

The source brightness distribution Iν(l,m) and the visibilities 𝒱ν(u,v,w) for an interferometer in three dimensions are related by

𝒱ν(u,v,w)=Iν(l,m)(1-l2-m2)1/2exp[-i2π(ul+vm+wn)]dldm. (\ref{eqn:3Dinterferometer})

For a two-dimensional interferometer confined to the (u,v) plane, the source brightness distribution Iν(l,m) is the Fourier transform of the fringe visibilities 𝒱ν(u,v):

Iν(l,m)(1-l2-m2)1/2=𝒱ν(u,v,0)exp[+i2π(ul+vm)]dudv. (\ref{eqn:3DSourceBrightness})

The point-source sensitivity (or brightness sensitivity in units of flux density per beam solid angle) for an interferometer with N antennas, each with effective area Ae, is

σS=2kTsAe[N(N-1)Δντ]1/2. (\ref{eqn:fluxsensitivity})

The brightness sensitivity (K) corresponding to a point source sensitivity σS and a beam solid angle ΩA is

σT=(σSΩA)λ22k, (\ref{eqn:brightsensitivity})

where ΩA=πθHPBW2/(4ln2)1.133θ02 for a Gaussian beam of HPBW θHPBW.

The (nonrelativistic) Maxwellian distribution of particle speeds v is

f(v)=4v2π(m2kT)3/2exp(-mv22kT). (\ref{eqn:MaxwellianDistribution})

The free–free emission coefficient is

jν=π2Z2e6neni4c3me2(2meπkT)1/2ln(bmaxbmin), (\ref{eqn:FFemcoefficient})


bminZe2mev2. (\ref{eqn:impactmin})

The free–free absorption coefficient is

κ=1ν2T3/2[Z2e6cneni12π(mek)3]π24ln(bmaxbmin). (\ref{eqn:FFabscoefficient})

At frequencies low enough that τ1, the Hii region becomes opaque, its spectrum approaches that of a blackbody with temperature T104 K, and the flux density varies as Sν2. At very high frequencies, τ1, the Hii region is nearly transparent, and

Sν2kTν2c2τ(ν)ν-0.1. (\ref{eqn:freefreetransparent})

On a log-log plot, the overall spectrum of a uniform Hii region has a break near the frequency at which τ1.

The emission measure of a plasma is defined by

EMpccm-6los(necm-3)2d(spc). (\ref{eqn:EmissionMeasure})

The free–free optical depth of a plasma is

τ3.28×10-7(T104K)-1.35(νGHz)-2.1(EMpccm-6). (\ref{eqn:TauFF})

The ionization rate QH of Lyman continuum photons produced per second required to maintain an Hii region is

(QHs-1)6.3×1052(T104K)-0.45(νGHz)0.1(Lν1020WHz-1), (\ref{eqn:LyAlphaRate})

where Lν is the free–free luminosity at any frequency ν high enough that τ(ν)1.

The magnetic force on a moving charge is

F=q(v×B)c. (\ref{eqn:MagneticForce})

The gyro frequency is defined by

ωGqBmc. (\ref{eqn:GyroFrequency})

The (nonrelativistic) electron gyro frequency in MHz is

(νGMHz)=2.8(Bgauss). (\ref{eqn:ElectronGyro})

The Lorentz transform is

x=γ(x+vt),y=y,z=z,t=γ(t+βx/c), (\ref{eqn:event1})
x=γ(x-vt),y=y,z=z,t=γ(t-βx/c), (\ref{eqn:event2})


βv/c (\ref{eqn:Beta})


γ(1-β2)-1/2 (\ref{eqn:Gamma})

is called the Lorentz factor. If (Δx,Δy,Δz,Δt) and (Δx,Δy,Δz,Δt) are the coordinate differences between two events, the differential form of the (linear) Lorentz transform is

Δx=γ(Δx+vΔt),Δy=Δy,Δz=Δz,Δt=γ(Δt+βΔx/c), (\ref{eqn:DiffLT})
Δx=γ(Δx-vΔt),Δy=Δy,Δz=Δz,Δt=γ(Δt-βΔx/c). (\ref{eqn:DiffLTPrime})

The Thomson cross section of an electron is defined by

σT8π3(e2mec2)2. (\ref{eqn:ThomsonArea})

Magnetic energy density is given by

UB=B28π. (\ref{eqn:Umag})

The synchrotron power of one electron is

P=2σTβ2γ2cUBsin2α. (\ref{eqn:Power})

Synchrotron power averaged over all pitch angles α is

P=43σTβ2γ2cUB. (\ref{eqn:AveragePower})

The synchrotron spectrum of a single electron is

P(ν)=3e3Bsinαmec2(ννc)ν/νcK5/3(η)dη, (\ref{eqn:Spectrum})

where K5/3 is a modified Bessel function and the critical frequency is

νc=32γ2νGsinαγ2νGE2B. (\ref{eqn:CriticalFrequency})

The observed energy distribution of cosmic-ray electrons in our Galaxy is roughly a power law:

n(E)dEKE-δdE, (\ref{eqn:CRSpectrum})

where n(E)dE is the number of electrons per unit volume with energies E to E+dE and δ5/2. The corresponding synchrotron emission coefficient is

jνB(δ+1)/2ν(1-δ)/2. (\ref{eqn:EmissionSpectrum})

The (negative sign convention) spectral index of both synchrotron radiation and inverse-Compton radiation is

α=δ-12. (\ref{eqn:SyncSpectralIndex})

The effective temperature of a relativistic electron emitting at frequency ν in magnetic field B is

(TeK)1.18×106(νHz)1/2(Bgauss)-1/2. (\ref{eqn:ElectronEffectiveTemp})

At a sufficiently low frequency ν,

Sνν-5/2 (\ref{eqn:SSASlope})


(Bgauss)1.4×1012(νHz)(TbK)-2. (\ref{eqn:SSABfield})

For a given synchrotron luminosity, the electron energy density is

UeB-3/2. (\ref{eqn:ElectronU})

The total energy density of both cosmic rays and magnetic fields is

U=(1+η)Ue+UB, (\ref{eqn:TotalU})

where η is the ion/electron energy ratio.

At minimum total energy, the ratio of particle to field energy is 1 (equipartition):

particleenergyfieldenergy=(1+η)UeUB=43. (\ref{eqn:MinimumE})

The minimum-energy magnetic field is

Bmin=[4.5(1+η)c12L]2/7R-6/7gauss (\ref{eqn:minEB})

and the corresponding total energy is

Emin(total)=c13[(1+η)L]4/7R9/7ergs. (\ref{eqn:minEtot})

The synchrotron lifetime is approximately

τc12B-3/2, (\ref{eqn:synclifeapprox})

where the functions c12 and c13 in Gaussian CGS units are plotted in Figures 5.10 and 5.11. Frequency limits νmin=107 Hz and νmax=1011 Hz are commonly used.

The Eddington limit for luminosity is

(LEL)3.3×104(MM). (\ref{eqn:Eddington})

The nonrelativistic Thomson-scattering power is

P=σTcUrad. (\ref{eqn:PScattered})

The relativistic Doppler equation is

ν=ν[γ(1+βcosθ)]. (\ref{eqn:Doppler})

The net inverse-Compton power emitted is

PIC=43σTcβ2γ2Urad. (\ref{eqn:ICPower})

The IC/synchrotron power ratio is

PICPsyn=UradUB. (\ref{eqn:PowerRatio})

The average frequency ν of upscattered photons having initial frequency ν0 is

νν0=43γ2. (\ref{eqn:ICFrequency})

The maximum rest-frame brightness temperature of an incoherent synchrotron source is limited by inverse-Compton scattering to

Tmax1012K. (\ref{eqn:Tmax})

The apparent transverse velocity of a moving source component is

β(apparent)=βsinθ1-βcosθ. (\ref{eqn:ApparentBeta})

For any β the angle θm that maximizes β(apparent) satisfies

cosθm=β (\ref{eqn:BetaThetamax})


sinθm=γ-1. (\ref{eqn:GammaThetamax})

The largest apparent transverse speed is

max[β(apparent)]=βγ. (\ref{eqn:ApparentBetamax})

The transverse Doppler shift (at θ=π/2) is

νν=γ-1. (\ref{eqn:TransverseDoppler})

The Doppler boosting for Doppler factor δν/ν is in the range

δ2+α<SS0<δ3+α. (\ref{eqn:Boosting})

Thermal and nonthermal radio luminosities of star-forming galaxies are

(LTWHz-1)5.5×1020(νGHz)-0.1[SFR(M>5M)Myr-1] (\ref{eqn:LThermal})


(LNTWHz-1)5.3×1021(νGHz)-0.8[SFR(M>5M)Myr-1]. (\ref{eqn:LNonthermal})

The minimum mean density of a pulsar with period P is

ρ>3πGP2. (\ref{eqn:MinDensity})

A rotating magnetic dipole radiates power

Prad=23(m¨)2c3. (\ref{eqn:MagneticLarmor})

The spin-down luminosity of a pulsar is

-E˙-dErotdt=-4π2IP˙P3. (\ref{eqn:SpindownLuminosity})

The minimum magnetic field strength of a pulsar is

(Bgauss)>3.2×1019(PP˙s)1/2. (\ref{eqn:Bmin})

The characteristic age of a pulsar is defined by

τP2P˙. (\ref{eqn:PulsarAge})

The braking index of a pulsar in terms of its observable period P and the first and second time derivatives is

n=2-PP¨P˙2. (\ref{eqn:BI})

At frequency ν the refractive index of a cold plasma is

μ=[1-(νpν)2]1/2, (\ref{eqn:refractiveindex})

where νp is the plasma frequency

νp=(e2neπme)1/28.97kHz(necm-3)1/2. (\ref{eqn:PlasmaFrequency})

The group velocity of pulses is

vgc(1-νp22ν2). (\ref{eqn:GroupVelocity})

The dispersion delay of a pulsar is

(tsec)4.149×103(DMpccm-3)(νMHz)-2, (\ref{eqn:DispersionDelay})


DM0dne𝑑l (\ref{eqn:DM})

in units of pc cm-3 is the dispersion measure of a pulsar at distance d.

The Bohr radius of a hydrogen atom is

an=n22mee20.53×10-8cmn2. (\ref{eqn:AtomicRadius})

The frequency of a recombination line is

ν=RMc[1n2-1(n+Δn)2],where  RMR(1+meM)-1. (\ref{eqn:RecombFrequency})

The approximate recombination line separation frequency Δνν(n)-ν(n+1) for n1 is

Δνν3n. (\ref{eqn:ApproxFrequency})

The spontaneous emission rate is

An+1,n64π6mee103c3h6n55.3×109(1n5)s-1. (\ref{eqn:SponRate})

The normalized Gaussian line profile is

ϕ(ν)=cν0(M2πkT)1/2exp[-Mc22kT(ν-ν0)2ν02], (\ref{eqn:GaussianProfile})


Δν=(8ln2kc2)1/2(TM)1/2ν0 (\ref{eqn:LineFWHM})


ϕ(ν0)=(ln2π)1/22Δν. (\ref{eqn:LinePeak})

Rate balance is given by

nUAUL+nUBULu¯=nLBLUu¯. (\ref{eqn:RateBalance})

The detailed balance equations connecting Einstein coefficients are

gLgUBLUBUL=1, (\ref{eqn:BLUBUL})
AULBUL=8πhν03c3. (\ref{eqn:AULBUL})

The spectral line radiative transfer equation is

dIνds=-(hν0c)(nLBLU-nUBUL)ϕ(ν)Iν+(hν04π)nUAULϕ(ν). (\ref{eqn:spectralradxfer})

The Boltzmann equation for a two-level system is

nUnL=gUgLexp(-hν0kT). (\ref{eqn:BoltzmannNUNL})

The line opacity coefficient in LTE is

κ=c28πν02gUgLnLAUL[1-exp(-hν0kT)]ϕ(ν). (\ref{eqn:LineOpacity})

The excitation temperature Tx is defined by

nUnLgUgLexp(-hν0kTx). (\ref{eqn:ExcitationTemp})

The recombination-line opacity coefficient is

κ(ν0)(ne2Te5/2Δν)(4πe6h3me3/2k5/2c)(ln22)1/2 (\ref{eqn:RecombOpacityCoefficient})

and the recombination line opacity is

τL1.92×103(TeK)-5/2(EMpccm-6)(ΔνkHz)-1. (\ref{eqn:LineOpacity2})

The recombination line brightness temperature is given by

TLTeτL1.92×103(TeK)-3/2(EMpccm-6)(ΔνkHz)-1. (\ref{eqn:LineTemp})

The recombination line/continuum ratio is

TLTC7.0×103(Δvkms-1)-1(νGHz)1.1(TeK)-1.15[1+N(He+)N(H+)]-1, (\ref{eqn:LCRatio})

where [1+N(He+)/N(H+)]1.08.

The electron temperature from the line/continuum ratio is

(TeK)[7.0×103(νGHz)1.1 1.08-1(Δvkms-1)-1(TCTL)]0.87. (\ref{eqn:ElectronTemp})

Quantization of angular momentum is given by

L=n. (\ref{eqn:AngMomQuant})

The angular momentum of a diatomic molecule is

L=mre2ω, (\ref{eqn:AngMom})


m(mAmBmA+mB) (\ref{eqn:ReducedMass})

is the reduced mass and re is the separation of the atoms with masses mA and mB.

The rotational energy levels of a diatomic molecule with moment of inertia I are

Erot=J(J+1)22I,J=0,1,2,. (\ref{eqn:EnergyQuant})

For a transition satisfying the selection rule

ΔJ=±1, (\ref{eqn:SelectionRule})

the line frequency is

ν=hJ4π2mre2. (\ref{eqn:Frequency})

The minimum temperature needed to excite the JJ-1 transition at frequency ν is

Tminνh(J+1)2k. (\ref{eqn:MinimumTemp})

The spontaneous emission coefficient is

AUL=64π43hc3νUL3|μUL|2, (\ref{eqn:EmissionCoef})


|μJJ-1|2=μ2J2J+1 (\ref{eqn:DipoleMoment})

and μ is the electric dipole moment of the molecule.

The critical density is

n*AULσv, (\ref{eqn:CriticalDensity})

where σ10-15cm-2 is the collision cross section and v105cms-1 is the typical H2 molecular velocity.

The CO-to-H2 conversion factor XCO in our Galaxy is

XCO=(2±0.6)×1020cm-2(Kkms-1)-1. (\ref{eqn:COconversionfactor})

The Hi hyperfine line frequency is

ν10=83gI(memp)α2(RMc)1420.405751MHz. (\ref{eqn:Frequency2})

The Hi hyperfine line emission coefficient is

A102.85×10-15s-1. (\ref{eqn:EmissionCoef2})

The Hi spin temperature Ts is defined by

n1n0g1g0exp(-hν10kTs), (\ref{eqn:SpinTemp})

where g1/g0 = 3.

The Hi line opacity coefficient is

κ(ν)3c232πA10nHν10hkTsϕ(ν). (\ref{eqn:OpacityCoef})

The hydrogen column density ηH is defined as the integral of density along the line of sight:

ηHlosnH(s)ds. (\ref{eqn:ColumnDensityDef})

If the Hi line is optically thin (τ1) then the Hi column density is

(ηHcm-2)1.82×1018[Tb(v)K]d(vkms-1). (\ref{eqn:ColumnDensityEq})

If τ1 the hydrogen mass of a galaxy is

(MHM)2.36×105(DMpc)2[S(v)Jy](dvkms-1). (\ref{eqn:HydrogenMass})

The total mass of a galaxy is

(MM)2.33×105(vrotkms-1)2(rkpc). (\ref{eqn:TotalMass})