Atmospheric extinction

• Suppose that Earth’s atmosphere has mass absorption coefficient at wavelength .

• If f0 is flux of incoming beam above atmosphere, then observed flux:

• Convert f/ f0 to a magnitude difference:

• wherez ds

dh

f =f0 exp − ρ(h)ds0

∞

∫ ⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟ wheρe ds =seczdh.

Δm = −2.5log( f / f0 )= aλ sec z

a =2.5 og10 e( ) ρ(h)dh0

∞

∫

Continuous extinction

• Produced by continuum scattering processes in Earth’s atmosphere (mainly Rayleigh+aerosols).

• Use calibration of ac() for site, together with expression just derived to get:

0

0.5

1

1.5

2

2.5

3

3.5

4

3000 5000 7000 9000 11000Wavelength (Å)

ac (Mag/airmass)

Vertical extinction at Roque de los MuchachosObservatory, La Palma

f = f0 ⋅10−0.4ac (λ )secz

Telluric-line extinction• Water, O2 other molecules produce

rotation/vibration bands composed of many saturated absorption lines.

• e.g. Near-IR spectrum of B3V star:

Telluric A band in spectrum of BS 3084 14/11/94 CTIO

00.20.40.60.8

11.21.41.6

7530 7550 7570 7590 7610 7630 7650 7670 7690Wavelength [Å]

Normalised flux

Individual lines almost resolved in this high-resolution spectrum

Unresolved telluric lines• At low/moderate resolution (Δ ≥ 1Å) each

spectral pixel holds several saturated lines with continuum gaps in between.

• How does extinction scale with sec z?

0

0.2

0.4

0.6

0.8

1

1.2

7530 7630Wavelength [Å]

Normalised flux

Note how unresolved lines no longer appear saturatedat low resolution

Same spectrum as before, degraded in wavelengthresolution by factor 11.

Curve of growth for telluric lines• Stellar flux is diminished by the equivalent width of the lines.• Telluric lines have saturated Voigt profiles due to thermal

Doppler+pressure-broadening.• On saturated part of curve of growth,

W ∝ secz

slope=

1

Log W

Log (sec z)

Optically thin

Doppler core

Lorentzian wings

W ∝ n(secz)

W ∝secz

slope=0.5

Saturated

EW ∝τ 1/ 2 ∝ (secz)1/ 2

Calibrating telluric-line extinction• Extinction relation in spectral regions affected by

saturated telluric lines is therefore:

• Calibrate aL() by measuring B stars (and flux-calibration standards) at several different airmasses (sec z).

• At each wavelength, fit a relation of the form:

• Do this by converting flux to magnitude and plotting versus log(sec z).

• Then fit slope B() and intercept log A0()

f = f0 ⋅10−0.4aL (λ) secz( )β where β=0.5– 0.6

A(, z)=A0()(secz)B()

Flux calibration• Observed monochromatic “OB” magnitude:

• Flux-calibrated monochromatic “AB” magnitude:

• where fn is the flux in erg cm–2 s–1 Hz–1. • Calibrate target spectrum by observing a flux

standard spectrum with known AB()• Apply line + continuum airmass corrections to both

stars.• Then fit polynomial to OB-AB() versus .

OB ≡−2.5ogcounτs Å–1 s–1( )

AB≡−2.5ogfn −48.60

Slit-loss corrections• If observing a variable object faint enough to

have a nearby non-variable comparison star:

• Rotate spectrograph slit to observe comparison and variable simultaneously.

• Observe comparison star once with wide slit.• Ratio narrow/wide calibrates slit losses vs :

Slitloss

Synthetic photometry• Monochromatic flux distributions:

• Unit of received flux density: – 1 Jansky (Jy)=10–26 W m–2 Hz–1

– mJy commonly used, esp. in radio astronomy.

fn ()dn =f()d n =c

dn =c2 d

⇒ fn ()=2

cf ()

Monochromatic magnitude systems• Palomar:• Hubble Space Telescope:• Conversion from AB magnitude to mJy:

• Conversion from ST to AB magnitude:

ST ≡−2.5ogf + 21.10ABn ≡−2.5ogfn −48.60

fn1 mJy

=10−0.4(AB−16.4)

ABn =ST −5og( / 5500Å)

AB = 16.4

ST = 16.4

fn

fn = 1 mJy

5500 Å

erg cm–2 s–1 Å–1

Broadband magnitudes• Conventional magnitudes are defined relative to

Vega = 0:

• Useful in early days before good absolute calibrations of stellar photometry were available.

• In the Johnson V band at 5500Å = 550 nm, Vega (m=0) produces

– ~108 photons m–2 s–1 nm–1 at 550 nm, or equivalently– ~1000 photons cm–2 s–1 Å–1 at 550 nm (Johnson V)

• Zero points of AB and ST magnitudes are chosen so that in V band, AB(V) = ST(V) = m(V).

m ≡−2.5ogf

fVega