fourth iram millimeter interferometry school 2004: the atmosphere 1 the atmosphere at mm wavelengths...

Fourth IRAM Millimeter Interferometry School 2004: The atmosphere 1

The atmosphere

at mm wavelengths

Jan Martin Winters

IRAM, Grenoble


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


ConstituentsSpecies 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 10-6 variable!


Simplistic Approach

The atmosphere is a highly complex and nonlinear system (weather forecast)

For our purpose we describe it as being

Static t = 0and 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


Atmospheric modelEquation of state

p = (/M) RT = pi

Hydrostatic equilibrium

dp / dz = g = p/ (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)


Standard atmosphere

Midlatitude winter

Midlatitude summer

US standard


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


Stability against convection: A rising air bubble will cool adiabatically

Temperature structure (adiabatic):

dq = cv dT + pdV = 0,

EOS pdV+Vdp = (R/M)dT = (cpcv)dT

dT/dz = g / cp = 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)


Radiative transfer (I)_________ = – I(r,n)dI(r,n)

dsoptical 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


Radiative transfer (II)

Define a brightness temperature:

2h3 1 22

c2 exp(h/kT) –1 c2In TE: I = B(T) = ______ ________________ = ____ kT

h/kT<<1

c2 1

2k 2Tb = ___ __ I

Brightness temperature

Motivation:


Radiative transfer (III)

_____ = _ Tb(s) + T(s)

dTb(s)

d

=> formal solution:

Tb(s) = Tb(0) e(0,s) + T(s´) e(s´,s) s´) ds´s

Isothermal medium (equivalent effective atmospheric temperature TAtm):

Tb(s) = Tb(0) e(0,s) + TAtm (1 e(0,s))source attenuation atmospheric emission (additional noise, increases system temperature)


Radiative transfer (IV)Plane wave, travelling in x direction:

E(x,t) = E0 exp { i (kx - t) }complex wave vector k = 2/ Nwith complex refractive index

N = n + i k=>

Imaginary part k determines attenuation (=4k/) (absorption)

Real part n determines phase velocity (n=c/vp) (refraction)

Relation to radiation intensity:

I0 = cE02/8S

where S is the Pointing vector


Absorption coefficient

0nℓcm-100

nℓnℓ{h0/kT}stimulated emission

e.g., collision broadening profile (complex van Vleck & Weisskopf)

000i0–i

0

i

Line profile (I)

000

00[

]00

)(

2ncollvrel ~ p


Line profile (II)Collision broadening profile (van Vleck & Weisskopf)

0


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 = d RT/Md, pV = V RT/MV, p = T RT/MT

with

p = pd + pV, T = d + V, MT = (___ ___ + ____ ___)-1

1 d 1 V

MdT MVT


Water vapor (II)Precipitable water vapor column pwv (usually given in mm):

(pwv =) w = __ ∫ V dz = __ V,0 hV

hV: water vapor scale height

The amount of pwv can be estimated from the temperature and the relative humidity RH:

V[g/m3] = pV MV / RT = 216.5 pV[mbar] / T[K]

RH[%] = pV / psat * 100, psat[mbar] ≈ 6 ( T[K] / 273 )18

w= 106 g/m3, hV =2000 m=>

w[mm] = 0.0952 * RH[%] * ( T[K] / 273 )17

e.g.: T = 280K, RH = 30% => w = 4.4mm

1w

1w


Water vapor (III)H2O

H2OO2

22GHz

60GHz 118GHz 183GHz 325GHz 380GHz

368GHz

O2

3 mm

1 mm

2 mm


Water vapor (IV) Phase delay – excess path

Real part n of complex refractive index:

kn = 2/n = 2nc2vpvp=c/nExtra time: t = 1/c ∫ (n-1) ds

Excess path length: L = ct = 10-6 ∫ N(s) dswith refractivity N = 106 (n-1)

Exact determination: compute n throughout the atmosphereApproximate treatment: empirical Smith-Weintraub equation:

N = 77.6 ___ + 64.8 ___ + 3.776 *105 ___ f()

L = Ld + LV = 231cm + 6.52 w[cm]

pd pV pV

T T T2

induced dipole permanent dipoleO2,N2 H2O H2O

Sea level, isothermal atmosphere at 280K


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


Water vapor (V)

PhD Thesis Martina Wiedner (1998)


To be continued ...

…tomorrow morning in the session about

Atmospheric phase

correction

fourth iram millimeter interferometry school 2004: the atmosphere 1 the atmosphere at mm wavelengths...

Documents

atmosphere slide

s s s e s

s s ds s slide

source function s

grenoble slide

degraded slide

martin winters iram

c reason