Abstract
By the early 1700’s, astronomers had generally accepted the idea that the stars were objects much like the Sun. A common assumption at the time was that each star was roughly as bright and as large as the Sun, but that we see them as faint and tiny points of light because they are very, very far away. The earliest measurements of parallax invalidated the hopeful notion that all stars were of about the same brightness. The parallax of 61 Cyg showed that it was significantly less luminous than the Sun: plugging the measured parallax and apparent magnitude of its components into the “distance modulus” equation shows that 61 Cyg A is about 1/6th and 61 Cyg B is about 1/12th of the Sun’s luminosity. The parallax of Vega showed that it was significantly more luminous than the Sun (about 40X solar luminosity). Your own measurements in Project 31 can be translated into an estimate that Barnard’s Star is 2000X less luminous than the Sun. So, astronomers could no longer blithely assume that faint stars were necessarily far away, or that bright stars were necessarily close. By the early 1800’s it was clear that stars spanned a very wide range of luminosity, size, and mass. This didn’t shake the astronomers confidence that “the Sun is a star and vice versa”, but it showed that the solar-stellar species encompassed a wide variation in individual characteristics. Despite its widespread acceptance, until the late 1800’s there was a surprising paucity of good evidence to validate the “Sun is a star’” paradigm.
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Notes
- 1.
By the way, the fact that a diffraction grating creates a spectrum was one of the strongest arguments that light was a wave phenomenon, rather than a stream of particles.
- 2.
Similar gratings with 200 lines/mm are sold by Paton Hawksley and Rainbow Optics. If you already have one of these, it can be used instead of the SA-100. However, if you’re going to buy a grating for this series of projects, the SA-100 will be the best choice for most set-ups.
- 3.
Alternatively, you can mount the grating to your camera lens by making an adapter from a donut of thin cardboard (e.g. the backing of a pad of paper). Make the outside diameter the size of your camera’s lens cover, and the inside hole diameter just large enough to screw (or squeeze) in your SA-100 grating. You will probably need to use a little tape to securely hold the grating to the cardboard donut, and a few pieces of tape around the outside to attach the cardboard donut to your camera’s lens housing. On many “autofocus” lenses, the cardboard donut will be attaching to the focus ring of the lens, so you’ll need to be careful to make a reliable attachment of the grating+donut without impairing the motion of the focus mechanism. This “home-made” approach of mating the grating to your camera is cheap, but it isn’t very robust and it doesn’t let you easily adjust the grating’s rotational orientation.
- 4.
- 5.
Refer back to Project 1 for a reminder of the location of the celestial pole and the path that your star will follow as it is carried along by the diurnal rotation of the celestial sphere. As you aim toward stars in different parts of the sky, you’ll need to re-orient the grating rotation to maintain the desired north–south spread of the spectrum.
- 6.
There is an “astronomical” modification that removes an internal filter from the camera. This modification is popular among amateur astronomers because it extends the camera’s spectral response far enough into the red to record the astronomically important Hα wavelength with good sensitivity.
- 7.
The SA-100 grating has 100 lines/mm, so the diffraction angle of visible light in the first-order is about 3.5 degrees. Therefore the small angle approximation is valid to better than about ±1% accuracy.
- 8.
For example, I used a 55 mm lens (f = 55 ∙ 10−3 m), a grating with 100 lines/mm (d = .01 ∙ 10−3 m), and a camera whose pixels are 5.7 μm in size (p = 5.7 ∙ 10−6 m). The scale factor according to Eq. 5.3 would be about K ≈ 10.4 ∙ 10−10 m/pixel = 10.4 Å/pixel.
- 9.
Why is it “B – R” instead of “R – B”? It would be mathematically legitimate to consider −2.5log (Ired/Iblue) as an “R – B” color index, but it just isn’t done in the company of astronomers. For both historical and practical reasons, they always define a color index in terms of the short wavelength magnitude minus the long wavelength magnitude. Part of the reason for this standardization is that by using it, everyone understands the convention of color indices; which is that if the color index gets smaller (or more negative) then the object is bluer, and if the color index gets larger (or less negative) then the object is redder.
- 10.
Note that in this, we have ignored the possibility (a near certainty) that atmospheric extinction T(λ) changes as a function of time, weather, and pointing direction. A proper astronomical evaluation of star colors would recognize this effect, and each night’s observing schedule would include a fairly extensive set of observations whose purpose would be to determine the atmospheric extinction and anchor the colors to standard stars with well-calibrated magnitudes in all spectral bands.
- 11.
It is common in physics classes to describe blackbody radiation by its energy flux versus wavelength. However, the camera’s sensor is a photon detector, so the values used here are the wavelengths of peak photon flux. Peak energy flux occurs at somewhat different wavelengths.
- 12.
RSpec is available from the developer at www.rspec-astro.com. As of this writing, the cost is about $100. A fully functional 30-day trial version can be downloaded at no cost. VSpec is a freeware package available at http://www.astrosurf.com/vdesnoux/. ISIS is a freeware package available at http://www.astrosurf.com/buil/isis/isis_en.htm
- 13.
As an example, the width of my ST-8XE imager chip is 13.8 mm, so the goal (from Eq. 5.8) is to position the grating between D ≈ 77–115 mm (≈3 to 4.5 inches). As things worked out, my set-up has D ≈ 100 mm, which has performed nicely. In most cases, the grating should not be on the same filter wheel that carries your photometric or color filters: the filter wheel carriage is likely to be too close to the focal plane to give a well-dispersed spectrum. If the nosepiece of your CCD imager is too short to provide sufficient grating-to-focal-plane separation, your astronomical supply shop can provide a threaded extender.
- 14.
Applying dark and flat frames is called “image reduction” or “image calibration”, almost interchangeably in the world of astro-imaging. I use “image reduction” here in order (hopefully) to avoid confusion with the subsequent process of calibrating your spectra by determining the scale factor K (Angstroms per pixel).
- 15.
Most astronomical image-processing software packages offer a method of scaling the dark frames, so that one set of darks can be used to reduce science image with different exposure durations. In order for this to work, you need both a set of darks and a set of bias frames (bias frames are dark frames with zero exposure duration). The rationale for scaling darks will probably become clear after studying the discussion of dark-signal in Appendix B. It is generally satisfactory, but the use of scaled darks is likely to leave some residual artifacts in the image, caused by either “hot” or “cold” pixels that can’t be handled correctly by the dark-scaling routine.
- 16.
You will also see the multiple-line calibration done with a quadratic fit, which might include the zero-order point. There are arguments in favor of both approaches. For initial experience with a slitless spectrograph, the linear calibration seems to be easier to understand.
- 17.
This approach to identifying the Hα line, which is so critical to calibrating your spectra, may at first seem to be either magical, ad-hoc, or simply bizarre (depending on your point of view). However, it turns up in a wonderful piece of astronomical history. When the first spectrum of a quasar was taken, the spectral lines (emission lines in that case) were completely mysterious, not matching up with any known substance. In a spark of insight, Dr. Maarten Schmidt realized that the ratios of their wavelengths followed the ratios of the Balmer series. This was the clue that allowed him to recognize that they were, indeed, the Balmer lines, but red-shifted by a hitherto unprecedented amount. You will see those red-shifted lines in Project 37.
- 18.
Astronomers soon devised a catchy phrase to help them remember this particular sequence of letters.
- 19.
Here is a worked example. The typical size of zero-order star images in my backyard observatory set-up is about 3–4 pixels FWHM. My scale factor is K = 8.86 Å/pixel. Thus, the finest spectral resolution that I can achieve is about 30 Å. It is possible to measure the wavelength of a deep, isolated, spectral line to a better accuracy by doing a centroid or curve-fit calculation on the signal profile, but the line itself will always appear as a rounded dip with FWHM ≈ 30 Å. If the spectral feature is actually composed of two lines, they won’t be resolved; they will appear as a single “blended” line. This smoothing effect caused by the size of the zero-order star image will tend to completely hide the weakest spectral features.
- 20.
How close is “about the same sightline orientation”? That depends on the accuracy that you are seeking, and the quality of the sky on the night and location you are using. The closer to the zenith both observations are made, the smaller the effect of atmospheric extinction will be; and the closer in time the measurements are made, the less risk there is of a change in sky transparency between the two observations. As a minimum, the standard star should be observed on the same night as the program star; the two sightlines should both be aimed at least 30 degrees above the horizon, and the elevation angle for the two sightlines should differ by no more than 30 degrees.
- 21.
This is available on-line at ftp://ftp.stsci.edu/cdbs/grid/pickles.
- 22.
Alternatively, you can mount the grating to your camera lens by making an adapter from a donut of thin cardboard (e.g. the backing of a pad of paper). Make the outside diameter the size of your camera’s lens cover, and the inside hole diameter just large enough to screw (or squeeze) in your SA-100 grating. You will probably need to use a little tape to securely hold the grating to the cardboard donut, and a few pieces of tape around the outside to attach the cardboard donut to your camera’s lens housing. On many “autofocus” lenses, the cardboard donut will be attaching to the focus ring of the lens, so you’ll need to be careful to make a reliable attachment of the grating+donut without impairing the focus mechanism’s motion. This home-made approach of mating your grating to your camera is cheap, but it isn’t very robust and it doesn’t let you easily adjust the grating’s rotational orientation (which you will need to do for the next project).
- 23.
- 24.
Note that “same age” and “same composition” are assumptions that we haven’t proven with any of the projects in this book, although they do turn out to be correct, in most cases. In some cases, measurements of the parallax or proper motions of cluster stars justify the assumption that the stars are related – they are at the same distance from Earth, and are moving along parallel paths. Also in some cases, spectroscopic measurements of individual stars in such a cluster can confirm that they have the same composition.
- 25.
It is possible to use a DSLR or a single-shot color imager for this project but I don’t recommend either of them, as they introduce complications to the imaging and data analysis procedures. Photometry with a color camera is tricky because you usually need to arrange for a precise and repeatable de-focus in order to assure that your star images span about a dozen pixels; this issue is discussed in Appendix B. DSLR cameras present a limited dynamic range (usually 12 bits) compared to astronomical CCD imagers (usually 16 bits). For most star clusters, your photometry will need to span a brightness range of about 6 magnitudes, with a minimum SNR of about 20:1 in the faintest stars. This dynamic range can (barely) be accommodated in the linear range of a 16-bit CCD. With a DSLR, you would need to use multiple exposure values to capture both the faint and the bright stars. Data analysis would then entail deciding star-by-star whether the “long” or “short” exposure data is most useful, then adjusting to compensate for the different exposure values. I’ve seen beautiful color-magnitude diagrams made from DSLR images, but they require better-than-novice skill at making and processing images.
- 26.
You will hear reference to two types of CCD sensors: “anti-blooming” and “non-anti-blooming”. Either type can be used for photometry, but you should be aware that they have different characteristic curves. The anti-blooming type, often a favorite of astro-imagers, has special circuitry that starts to gradually bleed charge out of the pixel “buckets” long before the bucket is full. The purpose of this is to reduce the bleeding of charge that creates the characteristic “spikes” on over-exposed star images. The impact of this circuitry on photometry is that the curve of ADU versus signal begins to depart from linearity at an ADU level that is significantly less than the full-well value. At ADU values above this first departure from linearity, the ADU follows a gradually bending curve; but any value above the linear range is not useful for photometry. The non-anti-blooming sensors are more popular among photometrists, because they do not have this special anti-blooming circuitry. The characteristic curve of a non-anti-blooming sensor tends to be linear almost up to the full-well value.
- 27.
For example, I used photometric B-V-R filters, with an ST-8XE imager. The exposure durations were 1 minute in R-band, 2 minutes in V-band, and 4 minutes in B-band.
- 28.
For my set-up, accounting for image download time, each three-filter image set took about 8 minutes, for a total of about 3.5 hours to accumulate 25 consecutive sets. So, I began the image sequence when the target was about 2 hours before the meridian and ended with it about 2 hours past the meridian.
- 29.
There are complications that aren’t included here, particularly: (a) the measurement must be made with a standard spectral response, and (b) your measurements of instrumental magnitudes, IMstar and IMVega, must either be made under identical atmospheric conditions and air mass or be corrected for atmospheric effects. There are well-defined methods for doing these corrections, but they add quite a bit of work (both additional observations, and additional calculations). Any standard reference on astronomical photometry will describe the method (see, for example, Henden & Kaitchuck, Astronomical Photometry).
- 30.
The derivation of this equation presents a good opportunity for you to use your high-school algebra and logarithms. Begin with the inverse-square law (Eq. 5.14), take the log of both sides, and then multiply both sides by −2.5. Use the definition of magnitude (m = −2.5log(S) to simplify the left hand side. Recall that log(x2) = 2log(x), that log(a/b) = log(a) – log(b), and that log(10) = 1 to simplify the right had side. With some re-arranging and simplifying you’ll get Eq. 5.15. Either side of this equation defines the “distance modulus” of a celestial object, which is usually given the symbol μ.
- 31.
American Association of Variable Star Observers
- 32.
Judging from the experience of the few amateur astronomers who have succeeded with this project, if you are using a telescope with an aperture smaller than 14 inches you’ll probably be restricted to using “clear” (unfiltered) images. The light loss of restricting the spectral range is likely to give unacceptably low signal level with a V- or R-band filter.
- 33.
You don’t need to be excessively fastidious about excluding stars from the sky annulus. All modern differential photometry software routines include an algorithm that checks for the presence of stars and eliminates the affected pixels from the calculation of the sky background. These algorithms are generally excellent, if you don’t over-stress them with too many or too-bright stars in the sky annulus.
- 34.
If necessary, refer back to Project 29 for an example of the expected shape of a Cepheid variable’s lightcurve.
- 35.
Considering the significance of this star to the history of cosmology, there are surprisingly few measurements of its period in the literature. In 1929, Hubble reported P = 31.39 days for this star, based on his plates made with the 60- and 100-inch telescopes at Mt Wilson spanning 18 years of observations. In 1950–1951, Professor Walter Baade used the newly commissioned 200-inch telescope at Mt. Palomar and determined P = 31.384 days. Observations taken by amateur astronomers in 2010 yielded a period estimate of P ≈ 31.4 days. Interestingly, it is now known that some Cepheids display slightly inconstant periods; but the record for this particular star is too sparse to demonstrate either constancy or variation in its pulsation period.
- 36.
Either side of this equation defines the distance modulus (μ) of a celestial object. The modern value for the M31 galaxy is μ = 24.47 ±0.06 mag.
- 37.
If the data points are not spaced at nearly equal intervals, and/or they don’t line up with an integral number of periods, then the standard practice is to translate the data into a phased lightcurve and perform a phase-weighted mean, using
$$ \left\langle m\right\rangle =-2.5 \log \left[{\displaystyle \sum_{i=1}^n\frac{\left({\phi}_{i+1}-{\phi}_{i-1}\right)}{2}}{10}^{-0.4{m}_i}\right] $$where ϕ i is the phase of the ith data point.
- 38.
RSpec is available from www.rspec-astro.com and currently costs about $100. A fully functional 30-day trial version can be downloaded at no cost. Vspec is freeware available at http://www.astrosurf.com/vdesnoux/. ISIS is freeware available at http://www.astrosurf.com/buil/isis/isis_en.htm
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Buchheim, R.K. (2015). Astrophysics and cosmology. In: Astronomical Discoveries You Can Make, Too!. Springer Praxis Books(). Springer, Cham. https://doi.org/10.1007/978-3-319-15660-6_5
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