The atmosphere at mm wavelengths

1. The atmosphere at mm wavelengths Jan Martin Winters IRAM, Grenoble Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

2. Why bother about the atmosphere?Because the atmosphere... • emits thermally and therefore adds noise • attenuates the incoming radiation • introduces a phase delay, i.e. it retards the incoming wave fronts • is turbulent, i.e. the phase errors are time dependent („seeing“) and lead to a decorrelation of the visibilities measured by an interferometer, i.e. the measured amplitudes are degraded Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

3. Constituents Species molec. weight Volume abundance amu N2 28 0.78084 O2 32 0.20948 Ar 40 0.00934 99.966% CO2 44 3.33 10-4 Ne 20.2 1.82 10-5 He 4 5.24 10-6 CH4 16 2.0 10-6 Kr 83.8 1.14 10-6 H2 2 5 10-7 => evaporated O3 48 4 10-7 N2O 44 2.7 10-7 H2O 18 a few 10-6variable! Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

4. Simplistic Approach The atmosphere is a highly complex and nonlinear system (weather forecast) For our purpose we describe it as being Static d / dt = 0and v = 0 1-dimensional f(r,f,q) -> f(z) Plane-parallel Dz / R << 1 In LocalThermodynamic Equilibrium (LTE) at temperatureT(z) Equation of state ideal gas Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

5. Atmospheric model Equation of state p = (r/M) RT = S pi Hydrostatic equilibrium dp / dz = -rg = - pM / (RT) g => dp / p = -gM / (RT) dz => p = p0 exp(-z/H) with the pressure scale height H = RT/gM (= 6 ... 8.5km for T=210 ... 290K) Temperature structure (tropospheric) dT/dz = -b (= 6.5 K/km) for z < 11 km T = T0 – b (z-z0) Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

6. US standard Standard atmosphere Midlatitude winter Midlatitude summer Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

7. Atmospheric structure: Stability (I) • Ground a) heats up faster than air during the day b) cools off faster than air during the night => Temperature gradient near the ground (< 2km) can be steeper or shallower than in the „standard atmosphere“ • Temperature inversion: e.g. if ground cools faster than the air, dT/dz > 0 usually stops abruptly at 1-2km altitude, normal gradient resumes Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

8. Atmospheric structure: Stability (II) • Stability against convection: A rising air bubble will cool adiabatically Temperature structure (adiabatic): dq = cv dT + pdV = 0, EOS => pdV+Vdp = (R/M)dT = (cp-cv)dT => dT/dz = -g / cp = -Gad(= adiabatic lapse rate = 9.8 K/km) If b > Gad, a rising bubble will become warmer than the surroundings (and less dense) => unstable (upward convection, e.g. if ground heats up faster than air) Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

9. dIn(r,n) ds dIn(s´) dtn _________ = en – knIn(r,n) _________ = – In(s´) + Sn(s´) Radiative transfer (I) optical depth: dtn = knds, source function Sn = en/kn => formal solution: s In(s) = In(0) e-tn(0,s) +  Sn(s´) e-tn(s´,s) kn(s´) ds´ 0 Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

10. c2 1 2k n2 Tb= ___ __ In Radiative transfer (II) Brightness temperature Motivation: 2hn3 1 2n2 c2 exp(hn/kT) –1 c2 In TE: In = Bn(T) = ______________________ = ____ kT hn/kT<<1 Define a brightness temperature: Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

11. s Tb(s) = Tb(0) e-tn(0,s) +  T(s´) e-tn(s´,s) kn(s´) ds´ 0 Radiative transfer (III) dTb(s) dtn _____= _Tb(s) + T(s) => formal solution: Isothermal medium (equivalent effective atmospheric temperature TAtm): Tb(s) = Tb(0) e-tn(0,s) + TAtm (1 - e-tn(0,s)) source attenuation atmospheric emission (additional noise, increases system temperature) Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

12. Radiative transfer (IV) Plane wave, travelling in x direction: E(x,t) = E0 exp { i (kx - t) } complex wave vector k = 2p/l N with complex refractive index N = n + i k => Imaginary partkdetermines attenuation (=4pk/l) (absorption) Real part n determines phase velocity (n=c/vp) (refraction) Relation to radiation intensity: I0 = cE02/8p (= |<S>T|) where S is the Pointing vector Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

13. n n0-nn0+n [ pn0(n0-n) 2 + Dn 2 (n0+n) 2 + Dn 2 Dn Dn ] ) ( (n0-n) 2 + Dn 2 (n0+n) 2 + Dn 2 Line profile (I) Absorption coefficient k0(n) = nℓ s(n) [cm-1] s0F0(n) (nℓ -> nℓ (1-{hn0/kT}), stimulated emission) e.g., collision broadening profile (complex van Vleck & Weisskopf) F0(n) = n / (pn0)[1/(n0-n- i Dn) + 1/(n0+n –i Dn)] F0(n) = ________________________ + ___________________ + i ___________________ + ___________________ Dn = 1/(2pt) = 1/(2p) n scollvrel ~ p Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

14. Line profile (II) Collision broadening profile (van Vleck & Weisskopf) Dn = 0.1n0 Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

15. 1 rd 1 rV Md rT MV rT Water vapor (I) • The amount of water vapor is highly variable in time (evaporation/condensation process) • => separate description in terms of „dry“ and „wet“ component (no clouds!) Partial pressures: dry wet total pd = rd RT/Md, pV = rV RT/MV, p = rT RT/MT with p =pd + pV, rT = rd + rV, MT = (______ + _______)-1 Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

16. 1 1 rw rw Water vapor (II) Precipitable water vapor column pwv (usually given in mm): (pwv =) w = __ ∫ rV dz = __rV,0 hV hV: water vapor scale height The amount of pwv can be estimated from the temperature and the relative humidity RH: rV[g/m3] = pV MV / RT = 216.5 pV[mbar] / T[K] RH[%]=pV /psat * 100, psat[mbar] ≈ 6 ( T[K] / 273 )18 rw=106 g/m3, hV =2000 m => w[mm] = 0.0952 * RH[%] *( T[K] / 273 )17 e.g.: T = 280K, RH = 30% => w = 4.4mm Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

17. 3 mm 2 mm 1 mm Water vapor (III) H2O 368GHz 22GHz O2 H2O O2 60GHz 118GHz 183GHz 325GHz 380GHz Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

18. pd pV pV induced dipole permanent dipole O2,N2 H2O H2O T T T2 Water vapor (IV) Phase delay – excess path Real part n of complex refractive index: kn = 2p/l n = 2pnn/c = 2pn/vp , vp=c/n Extra time: Dt = 1/c ∫ (n-1) ds Excess path length: L = cDt =10-6 ∫ N(s) ds with refractivity N = 106 (n-1) Exact determination: compute n throughout the atmosphere Approximate treatment: empirical Smith-Weintraub equation: N = 77.6 ___ + 64.8 ___ + 3.776 *105___ f(n) L = Ld + LV= 231cm + 6.52 w[cm] Sea level, isothermal atmosphere at 280K Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

19. Water vapor (V) • Atmosphere is turbulent • Water vapor is poorly mixed in dry air => „bubbles“ • These are blown by the wind across the interferometer array • => time dependent (fluctuating) amount of pwv along the line of sight in front of each telescope • => time variable phase variation, timescales seconds to hours Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

20. Water vapor (V) PhD Thesis Martina Wiedner (1998) Fourth IRAM Millimeter Interferometry School 2004: The atmosphere

21. To be continued ... …tomorrow morning in the session about Atmospheric phase correction Fourth IRAM Millimeter Interferometry School 2004: The atmosphere