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

2. 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

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

4. 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  t = 0  and v = 0 1-dimensional f(r,  )  f(z) Plane-parallel  z / R << 1 In Local Thermodynamic Equilibrium (LTE) at temperature T(z) Equation of state ideal gas

5. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 5 Atmospheric model Equation of state p = (  /M) RT =  p i Hydrostatic equilibrium dp / dz =  g =  p  / (RT) g  dp / p =  gM / (RT) dz  p = p 0 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 = T 0 – b (z-z 0 )

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

7. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 7 Atmospheric structure: Temperature Greenhouse effect Energy balance: 4  r 2    r 2  L sun /(4  R 2 ) (Albedo A = 0.33) BB emission = absorbed solar radiation => T = 252 K (=  21C) However, the average temperature near the ground is 288 K (= 15C) Reason: H 2 O, CO 2, CH 4, N 2 O absorb infrared radiation => energy is trapped in the atmosphere

8. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 8 Atmospheric transmission Radio cm mm sub-mm infrared optical UV

9. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 9 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

10. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 10 Stability against convection: A rising air bubble will cool adiabatically Temperature structure (adiabatic): dq = c v dT + pdV = 0, EOS  pdV+Vdp = (R/M)dT = (c p  c v )dT  dT/dz =  g / c p =  ad (= adiabatic lapse rate = 9.8 K/km) If b >  ad, a rising bubble will become warmer than the surroundings (and less dense) => unstable (upward convection, e.g. if ground heats up faster than air) Atmospheric structure: Stability (II)

11. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 11 Radiative transfer (I) _________ =  –  I (r,n) dI (r,n) ds optical depth: d  =  ds, source function S =  /  _________ = – I (s´) + S (s´) dI (s´) d  => formal solution: I (s) = I (0) e  (0,s) +  S (s´) e  (s´,s)   s´) ds´ s 

12. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 12 Radiative transfer (II) Define a brightness temperature: 2h 3 1 2 2 c 2 exp(h /kT) –1 c 2 In TE: I = B (T) = ______ ________________ = ____ kT h /kT<<1 c 2 1 2k 2 T b = ___ __ I Brightness temperature Motivation:

13. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 13 Radiative transfer (III) _____ = _ T b (s) + T(s) dT b (s) d  => formal solution: T b (s) = T b (0) e  (0,s) +  T(s´) e  (s´,s)   s´) ds´ s  Isothermal medium (equivalent effective atmospheric temperature T Atm ): T b (s) = T b (0) e  (0,s) + T Atm (1  e  (0,s) ) source attenuation atmospheric emission (additional noise, increases system temperature)

14. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 14 Radiative transfer (IV) Plane wave, travelling in x direction: E(x,t) = E 0 exp { i (kx -  t) } complex wave vector k = 2  / N with complex refractive index N = n + i k => Imaginary part k determines attenuation (  =4  k/ ) (absorption) Real part n determines phase velocity (n=c/v p ) (refraction) Relation to radiation intensity: I 0 = cE 0 2 /8  S    where S is the Pointing vector

15. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 15 Absorption coefficient  0  n ℓ  cm -1    0  0   n ℓ  n ℓ  {h 0 /kT}  stimulated emission  e.g., collision broadening profile (complex van Vleck & Weisskopf)  0  0    0    i  0  –  i   0        i    Line profile (I)  0   0      0      0   0  [ ]  0      0      ) (  2  n  coll  v rel ~ p

16. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 16 Line profile (II) Collision broadening profile (van Vleck & Weisskopf)  0

17. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 17 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 p d =  d RT/M d, p V =  V RT/M V, p =  T RT/M T with p = p d + p V,  T =  d +  V, M T = ( ___ ___ + ____ ___ ) -1 1  d 1  V M d  T M V  T

18. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 18 Water vapor (II) Precipitable water vapor column pwv (usually given in mm): (pwv =) w = __ ∫  V dz = __  V,0 h V h V : water vapor scale height The amount of pwv can be estimated from the temperature and the relative humidity RH:  V [g/m 3 ] = p V M V / RT = 216.5 p V [mbar] / T[K] RH[%] = p V / p sat * 100, p sat [mbar] ≈ 6 ( T[K] / 273 ) 18  w = 10 6 g/m 3, h V =2000 m => w[mm] = 0.0952 * RH[%] * ( T[K] / 273 ) 17 e.g.: T = 280K, RH = 30% => w = 4.4mm 1  w 1

19. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 19 Water vapor (III) H2OH2O H2OH2O O2O2 22GHz 60GHz118GHz183GHz 325GHz 380GHz 368GHz O2O2 3 mm 1 mm 2 mm

20. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 20 Water vapor (IV) Phase delay – excess path Real part n of complex refractive index: kn = 2  /  n = 2  n  c  2  v p  v p =c/n Extra time:  t = 1/c ∫ (n-1) ds Excess path length: L = c  t = 10 -6 ∫ N(s) ds with refractivity N = 10 6 (n-1) Exact determination: compute n throughout the atmosphere Approximate treatment: empirical Smith-Weintraub equation: N = 77.6 ___ + 64.8 ___ + 3.776 *10 5 ___ f( ) L = L d + L V = 231cm + 6.52 w[cm] p d p V p V T T T 2 induced dipole permanent dipole O 2,N 2 H 2 O H 2 O Sea level, isothermal atmosphere at 280K

21. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 21 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

22. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 22 Water vapor (V) PhD Thesis Martina Wiedner (1998)

23. Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 23 To be continued... …tomorrow morning in the session about Atmospheric phase correction