Practical applications: CCD spectroscopy

• Tracing path of 2-d spectrum across detector– Measuring position of spectrum on detector

– Fitting a polynomial to measured spectrum positions

• Optimal extraction of spectra from CCD images with simultaneous sky background subtraction:

– Scaling a profile + constant background

• Wavelength calibration of 1-d spectra– Measurement of positions of arc-line images

– Fitting a polynomial to measured positions of images of arc lines with known wavelengths

Observing hints• Rotate detector so that arc lines are parallel to

columns:

• To minimise slit losses due to differential refraction, rotate slit to “parallactic angle”

– i.e. keep it vertical:

• Spectra are then tilted or curved due to– camera distortions

– Differential refraction

To zenith

x

x

Night sky lines

Target spectrum

(Reference spectrum)

After bias subtraction and flat fielding

• Recall Lecture 3 for subtraction of B(x,), construction of flat field F(x,) and measurement of gain factor G.

• Corrected image values areD(x,) =[C(x, )−B(x,)] / F(x,)

=sky(x,) + f()P(x,)

Var(D) =Var(C) / F 2

= σ 02 +(C−B) /G[ ] / F 2

≈σ 02 + (C−B) /G[ ] sinceF ~1

Tracing the spectrum

• Spectra may be tilted, curved or S-distorted.

• Trace spectrum via a sequence of operations:

– divide into -blocks

– measure centroid of spectrum in each block (fit gaussian)

– fit polynomial in to calibrate x0().

• Once this is done, use x0() to select object/sky regions on subsequent steps.

x0

xx0

x

Sky subtraction• Alignment (rotation) of CCD detector

relative to grating aims to make ~const along columns.

• Imperfect alignment gives slow change in along columns.

• This causes gradient, curvature of sky background when is close to a night-sky line.

• Solution: fit low-order polynomial in x to sky background data.

• Alternative: fit linear function to interpolate sky from “sky regions” symmetric on either side of object spectrum:

on edge of night-sky emission line

away from night-sky emission line

x

TargetRefstar

Ref

Target

Slices across spectrum at =const:

“Normal” extraction

• Subtract sky fit, and sum the counts between object limits:

• Dilemma: How do we pick x1, x2?

– too wide: too much noise

– too narrow: lose counts

A = D(x)−S(x)( )x=x1

x2

∑

Var( A) = Var D(x)( )x=x1

x2

∑

S(x)

xx1 x2

Slice across spectrum at =const:

Optimal extraction

• 1) Scale profile to fit the data:

• 2) Compute σ(x) from the model:

• 3) σ-clip to “zap” cosmic-ray hits.

• Iterate 1 to 3, since σ(x) depends on A:

S(x)

xx1 x2

Slice across spectrum at =const: Starlight

profile P(x)D(x) =S(x) +AP(x)

ˆ A=D(x) −S(x)( )P(x) / σ 2 (x)

x∑P2 (x) /σ 2 (x)

x∑

Var ( ˆ A ) =1

P2 (x) / σ 2 (x)x∑

σ 2(x) =σ02(x)+

S(x)+ ˆ A P(x)G

Estimating the profile P(x)• The fraction of the

starlight that falls in row x varies along the spectrum and is given by:

• This is an unbiased but noisy estimator of P(x).

• It varies as a slow function of wavelength.

• Plot against and fit polynomials in at each x.

D−S(D−S)

x∑-0.05

0

0.050.1

0.150.2

0.250.3

0.350.4

0.45

0 20 40 60 80

Column ( )

Fraction of starlight in

row

2 rows below

1 row below

Centre row

1 row above

2 rows above

3 rows above

00.05

0.10.15

0.2

0.250.3

0.350.4

-3 -2 -1 0 1 2 30

0.050.1

0.150.2

0.250.3

0.350.4

-3 -2 -1 0 1 2 3

Column 20 Column 60

Optimal vs. normal extraction

• Pros:– Optimal extraction gives lower statistical noise.

– Equivalent to longer exposure time

– Incorporates cosmic-ray rejection

• Cons:– Requires P(x,) slowly varying in (point sources).

• Essential papers:– Horne, K., 1986. PASP 98, 609

– Marsh, T. and Horne, K.

Wavelength calibration

• Select lines using peak threshold.

• Measure pixel centroid xi by computing x or fitting a gaussian

• Identify wavelengths i

• Fit polynomial (x) to i, i=1,...,N.

• Reject outliers (usually close blends)

• Adjust order of polynomial to follow structure without too much “wiggling”.

-100

0

100

200

300

400

500

0 20 40 60 80

threshold level

Dealing with flexure

• Flexure of spectrograph causes position xi of a given wavelength to drift with time.

• Measure new arcs at every new telescope position.

• Interpolate arcs taken every 1/2 to 1 hour when observing at same position.

• Master arcfit: Use a long-exposure arc (or sum of many short arcs) to measure faint lines and fit high-order polynomial.

• Then during night take short arcs to “tweak” the low-order polynomial coefficients.

Statistical issues raised

• Outlier rejection: what causes outliers, and how do we deal with them?

– Robust statistics.

• Polynomial fitting: how many polynomial terms should we use?

– Too few will under-fit the data.

– Too many can introduce “flailing” at ends of range.

• We’ll deal with these issues in the next lecture.