AST 325/6 LAB#2

Lab #2 “Introduction to Spectroscopy”

Of all objects, the planets are those which appear to us under the least varied aspect. We see how we may

determine their forms, their distances, their bulk, and their motions, but we can never known anything of

their chemical or mineralogical structure; and, much less, that of organized beings living on their surface…

Auguste Comte, The Positive Philosophy, Book II, Chapter 1 (1842)

1 Overview

Spectroscopy is a fundamental tool used by all physical sciences. For astrophysicists,

spectroscopy is essential for characterizing the physical nature of celestial objects and the

universe. Astronomical spectroscopy has been used to measure the chemical composition and

physical conditions (temperature, pressure, and magnetic field strength) in planets, stars, and

galaxies. Characterizing a spectrograph’s instrumental parameters is a key for deriving the

intrinsic spectra from a source. In this lab you will use a spectrograph to collect data from

common light sources (room lights and gas discharge lamps), establish a wavelength scale,

investigate the noise properties of the detector, and (hopefully) measure astronomical spectra of

stars and planets. In class we will explore the fundamentals of diffraction, dispersion elements,

2D detectors (e.g, CCDs), telescopes, and review the celestial coordinate systems.

2 Schedule

This is a four-week lab between October 7 (Monday) and November 6 (Wednesday). There will

be no class on October 14 and November 4 because of Thanksgiving and Reading Week,

respectively. Group-led discussions will happen on October 21 and 28. Starting the week of

October 21, we will attempt to obtain astronomical spectra using the campus telescope. The

details of the Campus Telescope Sessions will be announced later in the class and on its web

page.

3 Goals

Use a simple, visible light (350−700 nm) spectrometer to explore the spectra of laboratory

and astrophysical sources. We will conduct wavelength calibration of the Ocean Optics

spectrometer, which will introduce the concept of spectroscopy and linear least square fitting.

We will then learn how to acquire data from a telescope and acquire spectra of astronomical

sources. In the process, we will learn the basic steps of astronomical data reduction

(e.g., dark subtraction, flat fielding, wavelength solution) using the SBIG spectrograph on the

MP telescope.

3.1 Reading assignments

● USB 2000 spectrometer handout (class web page)

● Lecture/notes on statistics and error analysis and least square fitting (class web page)

● Notes on CCDs and their noise properties (class web page)

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● Reference: Chapters 1−4 “Handbook of CCD Astronomy,” S. B. Howell, Cambridge

University Press. There are two copies in Gerstein and more copies may be available in

AB 105. Pay special attention to §§3.4−3.8 and §§4.2−4.3, & 4.5.

● Reference: “To Measure the Sky,” F. R. Chromey, Cambridge University Press. In

particular, Chapters 6 on Astronomical Telescopes, Chapter 8 on Detectors, and Chapter

11 on Spectrometers.

4 Key Steps

1. Learn to operate the USB 2000 spectrometer using the Spectral Suite software in one of

the lab tutorials during the first week of the lab.

2. Save spectra and read them into Python for plotting and analysis.

3. Observe and compare spectra of common sources—incandescent lamp, fluorescent strip

light, gas discharge lamps, and sunlight.

4. Determine the wavelength calibration of the spectrometer, i.e., the mapping between

pixel number and wavelength. Do this by measuring the centroids (i.e., pixel positions) of

bright Neon (and Hydrogen, if available) lines with respect to their known wavelength.

Then use the method of linear least squares to determine a polynomial fit to these data to

derive the wavelength solution.

5. Attend one of the telescope sessions and use the SBIG spectrometer at the campus

observatory to collect spectra of the moon, Arcturus, Vega, and/or Jupiter and other

bright stars.

6. Reduce the SBIC spectra of the astronomical sources and compare and contrast these

spectra.

7. Write up your report.

5 Linear Least Squares Fitting

One of the primary skills we will learn in this lab is the use of linear least squares fitting. Often

observations and experimental measurements are undetermined, which means that they are

limited by the number of observations or sampling to calculate an undetermined parameter space.

To correct for this, we use an equation to model a set of data, and compare the difference

between the observed values to the fitted values from the model. This difference is referred to as

residuals. The term “least-squares” refers to minimizing the square of the residuals to determine

the best-fit model to the observed data set.

In this lab, we will first focus on linear-least squares where the model is a straight line, but we

will generalize the least squares method to other non-linear functions. It is important that in this

lab you do not use a canned least-squares routine and you write your own least-square

routine.

5.1 A Straight Line Fit

Suppose that we have a set of N observations (xi, yi) where we believe that the measured value, y,

depends linearly on x, i.e.,

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For example, suppose a body is moving with constant velocity, what is the speed (m) and initial

(c) position of the object?

Given our data, what is the best estimate of m and c? Assume that the independent variable, xi, is

known exactly, and the dependent variable, yi, is drawn from a Gaussian probability distribution

function with constant standard deviation i = const. Under these circumstances the most likely

values of m and c are those corresponding to the straight line with the total minimum square

deviation, i.e., the quantity

is minimized when m and c have their most likely values. Figure 1 shows a typical deviation.

The best values of m and c are found by solving the simultaneous equations,

Evaluating the derivatives yields

This set of equations can conveniently be expressed compactly in matrix form,

and then solved by multiplying both sides by the inverse,

The inverse can be computed analytically, or in Python it is trivial to compute the inverse

numerically, as follows.

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Figure 1: Example data with a least squares fit to a straight line. A typical deviation from the straight line is

illustrated.

5.2 Example Python Script

# Test least squares fitting by simulating some data.

import numpy as np

import matplotlib.pyplot as plt

nx = 20 # Number of data points

m = 1.0 # Gradient

c = 0.0 # Intercept

x = np.arange(nx,dtype=float) # Independent variable

y = m * x + c # dependent variable

# Generate Gaussian errors

sigma = 1.0 # Measurement error

np.random.seed(1) # init random no. generator

errors = sigma*np.random.randn(nx) # Gaussian distributed errors

ye = y + errors # Add the noise

plt.plot(x,ye,’o’,label=’data’)

plt.xlabel(‘x’)

plt.ylabel(‘y’)

# Construct the matrices

ma = np.array([ [np.sum(x**2), np.sum(x)],[np.sum(x), nx ] ] )

mc = np.array([ [np.sum(x*ye)],[np.sum(ye)]])

# Compute the gradient and intercept

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mai = np.linalg.inv(ma)

print ‘Test matrix inversion gives identity’,np.dot(mai,ma)

md = np.dot(mai,mc) # matrix multiply is dot

# Overplot the best fit

mfit = md[0,0]

cfit = md[1,0]

plt.plot(x, mfit*x + cfit)

plt.axis(‘scaled’)

plt.text(5,15,’m = {:.3f}\nc = {:.3f}’.format(mfit,cfit))

plt.savefig(‘lsq1.png’)

See Figure 2 for the output of this program.

Figure 2—Least squares straight line fit. The true values are m = 1 and c = 0.

5.3 Error Propagation

What are the uncertainties in the slope and the intercept? To begin the process of error

propagation we need the inverse matrix

so that we can compute analytic expressions for m and c,

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.

The analysis of error propagation shows that if z = z(x1, x2, .. xN) and the individual

measurements xi are uncorrelated (they have zero covariance) then the standard deviation of the

quantity z is

.

If the data points were correlated then we would have a covariance matrix. The diagonal

elements of this matrix are the standard deviations σii2 and of the off diagonal elements σij2 =

‹xixj›-‹xi›‹xj›.

Thus, ignoring any possible covariance

The expression for the derivative of the gradient, m, is

because ., where δ is the Kroneker If we assume that the measurement error is the

same for each measurement then

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Similarly, for the intercept, c,

and hence

If we do not know standard deviation, a priori, the best estimate is derived from the

deviations from the fit, i.e.,

Previously, when we compute the standard deviation the mean is unknown and we have to

estimate it from the data themselves; hence, the Bessel correction factor of 1/(N-1), because there

are N-1 degrees of freedom. In the case of the straight line fit there are two unknowns and there

are N-2 degrees of freedom.

6 Night time observing

For nighttime astronomy, we will use a spectrograph that is located on the 16-inch telescope on

the 16th floor of the McLennan Physics tower. Michael Williams (williams@astro.utoronto.ca),

who is in charge of the lab equipment and telescope operation, will be there to set up the

telescope and help you collect your data. Separate documents describing this spectrograph and

the observing procedure are available on the class web page.

Example spectra taken with the SBIG spectrograph attached to the 16-inch telescope are shown

in Figure 3. The top spectrum is for a quartz halogen lamp, and shows the response of the

spectrometer to an approximately 3200 K black body. (In this example, the short wavelength flux

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from the lamp may be suppressed by a built-in UV filter, so the blackbody assumption may not be

valid in the blue part of the spectrum.) Note the overall variation of responsivity and fine scale

pixel-to-pixel fluctuations. The subsequent astronomical spectra are corrected for the

spectrometer response assuming that the lamp radiates like a black body with temperature equal

to the color temperature. Thus we compute for each pixel, Pi, the quantity

where Ri is the raw signal, Di is the dark count, and Li is the lamp, and Bv(T) is the Planck

function

where νi = c /λi is the frequency of the i-th pixel. (Note that in this case the lamp spectrum is used

for flat fielding see lecture slides).

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Figure 3. Spectra of a lamp and some astronomical sources. Comparison of Arcturus (4300 K)

and the Sun (5800 K) shows the effect of Wien’s law. The stellar spectra show both the

underlying continuum emission and many microscale absorption lines. In the solar spectrum Ca

II H&K 393.37, 396.85 nm, the G band 430.8 nm, Hβ 486.1 nm, the b and E bands (Mg + Fe) 517,

527 nm, Na D 588.995, 589.592 nm, and Hα 656.2 nm are all visible. The spectrum of Jupiter is

red, with strong methane absorption at 619 nm. The exposure times are: lamp 23 ms, 1000

frames; Arcturus & Jupiter 500 ms, 100 frames; sun 3 ms, 100 frames. The astronomical spectra

are dark subtracted, divided by the lamp spectrum, and multiplied by a 3200 K black body.

AST 325/6

# Lab #2 “Introduction to Spectroscopy”

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