|I have looked farther into space than ever [a] human being did before me.|
|--William Herschel, c. 1780|
There are several different concepts that work together in this lab to finally give us the answer we are looking for, the distance to the Pleiades. (The Pleiades is a star cluster, i.e., a group of stars located near one another in space that were probably 'born' at the same time). The technique you'll be using is called spectroscopic parallax, and is used to determine the distance to young star clusters in our galaxy but more than 10 pc from Earth.
The first step is to learn how to do a rough classification of stellar spectra. Although Fraunhofer first observed stellar spectra in the early 19th century, it was the turn of the twentieth century before Angelo Secchi and E.C. Pickering began using their spectra to classify stars. The classification system we use today is based on the one first proposed by Willamina Flemming (who started out as Pickering’s maid!) and refined by Annie Jump Cannon and Antonia Maury. Cannon and Maury’s work were published as the first Henry Draper Catalogue. The technique you will use is very similar to the techniques of these women. Today of course, astronomers can use computers to analyze the light from stars in a much more quantitative manner, allowing more refinement (and more certainty) than the technique you will use, although the basics are still the same. The odd order of letters is because Flemming’s system put stars in order by the strength of their hydrogen lines: A had the strongest broadest lines, B weaker and thinner, C somewhat weaker still, etc. Cannon rearranged and combined groups to reflect temperature: O are the hottest, B a little less hot, etc. (A question you might want to ponder, why would the medium-hot stars have the strongest Balmer lines?)
The second step is to find the brightness of this star. Today, astronomers use a device called a photometer (i.e. a photon counter) to measure the flux received from an individual star. Data are taken over a period of several hours through several different filters, and are usually combined with data taken over the course of several years. The European Space Agency’s Hipparcos mission made such “photometric” measurements of more than 120,000 stars during its 3 year mission. However, we need to fit this into a regular lab period, so again you'll be using a more historical technique: estimating the brightness based on the size of a photographic image.
With the spectral type and brightness, you can make a Hertzsprung-Russel (H-R) diagram. For stars on the main sequence, there is a relationship between the temperature and luminosity. Using Cannon's classification system, there is a relationship between the spectral type and the temperature. So, all main sequence stars of the same type have the same luminosity: you can determine the luminosity by knowing the spectral type. We use a set of standard stars at known distances (how do we know the distances?) to calibrate our luminosity scale.
The flux or apparent brightness is the amount of light per unit area received from a star, or the total amount of light the star puts out (luminosity) divided by the total area that light shines though. Since the light is emitted in all directions, the area will be the surface area of a sphere with a radius equal to the distance to the star. For a star at distance d:
Thus if we can determine the flux and luminosity of a star, we can determine its distance.
A note on Magnitudes
Historically, astronomers used magnitudes to talk about the brightness of stars. The first magnitude system was developed by the Greek astronomer Hipparchus, who divided the stars up into 6 groups, calling the brightest group the first magnitude, the second brightest the second magnitude, etc. Astronomers have refined the system, and now include negatives and numbers larger than 6. For example, the Sun has an apparent magnitude (m) of -26, Sirius a -1.42, and Hubble can see down to about 30. Since the human eye responds to light on a logarithmic scale (i.e. a light that actually puts out 10 times more photons than another light will only look about 2.5 brighter), the magnitude system is also logarithmic. Thus, the apparent magnitude system corresponds to the flux, but the numbers are in reverse order and a difference of 2.5 in magnitude corresponds to a factor of 10 in flux. Similarly, the absolute magnitude (M) corresponds to the luminosity, but again a bigger number is dimmer and a difference of 2.5 is equal to a factor of 10 in luminosity. By converting flux and luminosity into apparent and absolute magnitude, astronomers are able to derive the distance modulus:
where d is the distance to the star in parsecs. Since m is fairly straightforward to measure, especially from space, but M is less certain (where are all the possible errors?), astronomers sometimes prefer to use the distance modulus (m-M) rather than calculating the distance.
The spectral type of a star is just a letter and a number that designates what kind of a star it is. From hottest to coolest, the letter categories are O, B, A. F, G, K, M. The differences between spectral types show up in the absorption lines of the spectra of stars. Some astronomers spend their whole careers determining detailed spectral types, but in this lab, you're just going to make a rough estimate for ten stars in the Pleiades star cluster.
In lab, you will be given a large envelope by your instructor which should contain:
Lay out the standard star spectra sheets in front of you, so that you can see the whole sequence of stars from O to M. These are your reference spectra. Notice the sequence of letters is subdivided by numbers, which go from 0-9 for each letter. The wavelength scales on the standard spectra are all consistent with each other, so you can follow absorption lines up and down the page between spectral types. The Pleiades spectra are also consistent with themselves (notice how the few labelled lines follow in the same place from spectrum to spectrum), however they are not on the same scale as the standard spectra. The absorption lines in the Pleiades spectra are more widely separated than the standard star spectra. Although this makes life a little more difficult, you'll find it doesn't pose a major problem for classification.
Begin by using the flow chart, which asks you to make choices about labelled absorption lines in the particular spectra you are trying to type. Look at one Pleiades spectra one at a time and follow the flow chart. Try to answer as accuratly as possible, and only choose maybe if you really can't decide between yes or no. At the end, you are given a range of spectral types that your star could be, for example F2-F9.
Now go to the standard star spectra sheets and look at this range of spectral types (F2-F9 in our example). Look at how particular spectral lines (like the Hydrogen lines or Calcium lines) change their relative intensities through this range of spectral types. Now you must pick some of these lines and decide where your Pleiades spectrum fits in amongst the standard star spectra. You must use the relative intensities of two different lines (in the standard and the Pleiades) to decide this, and not the intensity of a single Pleiades line versus a single standard line because the brightness of the Pleiades spectra is not the same as the brightness of the standards.
Write which absorption lines you used to decide the spectral type in the Table 1. Particularly, include the lines which helped you make the final determination between the numerical subdivisions (note the absence of a line can be as important as its presence, so if you use the fact that a line is missing, include it in your list.) Also record the spectral types for each star. You're done with the spectral typing now (and also most of the work in the lab); congratulations!
On the back of the flowchart is a negative photograph of these ten stars. A brighter star makes a bigger 'dot' on this photograph, so if we knew how to turn the dot-size into a flux, we could just measure the sizes of the star-images on the photograph to find out their apparent magnitudes. Luckily, there is a table in this lab that gives you a calibration curve between star-image diameter (dot-size) and flux or apparent blue magnitude.
Measure the diameter of each of the ten stars (numbered on the photograph) to the nearest 0.1 mm. Then look at the table and find your measurement, then record the flux that corresponds to that measurement in the data table. If you are going to do part 4, it will save you time to also record the aparent magnitude now.
You have the information you need to make an H-R diagram for these ten Pleiades stars. Plot the flux vs. Spectral Type either on log graph paper or using excel (convert the spectral types to numbers: 0-9 for type O0 - O9, 10 - 19 for type B0-B9, etc.) You should find that the stars fall fairly close to a main sequence shape (a nearly straight line on a log graph). Now plot all the stars listed in the standard star main sequence table in this lab, using a different symbol than the one you used to plot the Pleiades stars. These stars are all main sequence stars that have known spectral types, luminosities and fluxes (how do they know the Luminosity?). These stars should fall on a very well-defined main sequence that is displaced vertically from your data points.
A best-fit line is a smooth line or curve that goes through or near as many points as possible. Its shape should match the shape the data appear to have. It is NOT a jagged line that goes from point-to-point (don't just connect the dots). Draw a 'best-fit' line through the standard star points from the table (a "trendline" if you're using excel). This shows you what a main sequence line should look like. Now draw a best-fit line through the points from the Pleiades data that you gathered. Hopefully they're somewhat similar. (If not, consider where you think your errors probably are). Now you're ready for the final step.
The Pleiades is a young star cluster (how can we tell that?), so all the stars in it are still on the main sequence. The standard stars are, of course, also main sequence stars. Since all main sequence stars with the same spectral type should have the same luminosity, we can use the graph to determine what the luminosity of the Pleiades stars should be by looking at the offset.
To determine the distance you will need the average value of L/ f. For several points on the graph (e.g. O0, B5, B0, etc) determine the luminosity for that type from the standard stars line, and flux from the pleiades line. Use points on your line, not data points. If you have the equations of the best-fit lines, you can put the spectral type in for the x value and calculate the luminosity or flux.
Create a table either in excel or on the back of your graph. Include the spectral type, the luminosity, the flux and L/f in the table. Also, make sure you include units. Once you have L, f and L/f for AT LEAST 8 values covering the full range of your data, find the average of L/f and record it on the worksheet.
Solve the equation for the definition of flux for distance. Show your work and the final equation on the worksheet.
Use the average value of L/ f to calculate the distance to the Pleiades. Show your work and record your final answer on the worksheet. Your final answer should be in the most appropriate distance unit (AU, pc, kpc, Mpc). This is a nearby star cluster, so check your answer against distances like the radius of the galaxy and the distance to the nearest star to see if it makes sense. You are expected to be within an order of magnitude (1 power of 10).
Answer the questions on the worksheet.
The standards and calibration curve tables also include magnitudes. Look up the apparent magnitudes for each of the stars in the Pleiades and record them the last column of table 1 on the worksheet. Make a new graph of the Pleiades and the standard stars using M and m on the vertical axis instead of L and f.
Your graph will again have two main sequence lines, one for the standard stars and one for the Pleiades. Draw a best-fit line for each data set. This time, you need to know the distance modulus (m-M) to get the distance. For at least 8 points on your best-fit line, find m and M (cirlce the points on your graph or have excel do these calculations if you have the equation for the best fit line.) For each spectral type, find (m-M) then find the average value of (m-M). Record the spectral type, m, M and the distance modulus in a table either in excel or on the back of your graph.
Solving the distance modulus for distance gives
Calculate the distance from the distance modulus. Show your work and your final answer on the worksheet. Make sure your units are the same as in the calculation from L and f.
Answer the questions that follow.
Last update 2/21/06
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