Download Project 9: Stellar Spectra: Temperature

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PROJECT 9 STELLAR SPECTRA: TEMPERATURE Objective: The main objective of this project is to show the students how to estimate the temperature of a star. Two different methods are shown. First, an estimate of the effective temperature of a star can be done by applying Wien's law. This method simply takes into account the wavelength at which the spectrum exhibits maximum flux. The second method consists of measuring the equivalent width of some spectral lines that are sensitive to changes in temperature. The lines used in this project are the hydrogen Hα line at 6563 Å and the doublet Na I lines at 5890 Å and 5896 Å. In this project the student can also learn how to extract information from the shape of a spectral line. The Equivalent Width and Full Width at Half Maximum are defined and their relationship with some physical properties of the stars and the instrumental set‐up used to obtain the spectrum is explained. Observations & Material: In this project the student will make use of real astronomical optical spectra of various spectral types obtained from the 1.3m telescope of the Skinakas Observatory (Crete, Greece). The measurement of line profile parameters is carried out with the package SPLOT of IRAF. SPLOT implements graphically all the techniques described in this project. In this way the students can visualize the processes that they are performing and understand their results. The students will be provided with • Low and intermediate resolution spectra of MK standards of various spectral types • Calibration spectra (ARC line spectra) • Spectra of various stars for temperature determination PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Contents: Stellar Spectra: Temperature 1. The Electromagnetic Spectrum 2. Stellar Spectra 2.1. Components of the stellar spectrum 2.2. Characterization of the spectral line profile 3. Stellar temperature 3.1 Wien’s Law 3.2 Spectral lines NOTE: Data used in this project can be found in the file “temperature.tar.gz” already downloaded with the “.zip” file. Stellar Spectra: Temperature 1. The Electromagnetic Spectrum The electromagnetic spectrum is the range of all possible electromagnetic radiation. The electromagnetic spectrum extends from below the frequencies used for modern radio (at the long‐wavelength end) through gamma radiation (at the short‐wavelength end), covering wavelengths from thousands of kilometers down to a fraction the size of an atom. Figure 1: Electromagnetic spectrum. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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In Astrophysics a spectrum is the display of stellar radiation as a function of wavelength (or energy). Astrophysicists treat light as a wave or a particle depending on the region of the electromagnetic spectrum they are interested in. It turns out that low‐energy photons (i.e. radio) tend to behave more like waves, while higher energy photons (i.e. X‐rays) behave more like particles. This is an important difference because it affects the quantities, hence the units, used to characterise the properties of light. For example, X‐rays and gamma‐rays are usually described in terms of energy (eV), optical and infrared light in terms of wavelength (microns or Angstroms), and radio in terms of frequency (Hz). It would be inconvenient to describe both low energy radio waves and high energy gamma‐rays with the same units because the difference in their energies is too great. A radio wave can have an energy on the order of 4 x 10‐10 eV as opposed to 4 x 109 eV for gamma‐rays. That is an energy difference of 1019, or ten million trillion eV! However, these quantities are related through the well known equations E=hν & λ=c/ν ‐27
where h=6.626 x 10 erg s is the Planck's constant, c=2.998 x 1010 cm/s is the speed of light, λ and ν the wavelength and frequency of the electromagnetic wave (light as waves) and E the energy of the photon (light like particles) Figure 2: An electromagnetic wave. The RXTE Learning
Center.
PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Exercise 1: Listed below are the approximate wavelength, frequency, and energy limits of the various regions of the electromagnetic spectrum. Some information is missing. Complete the table. Wavelength (m) Frequency (Hz) Energy (J) Radio > 1 x 10‐1 < 3 x 109 < 2 x 10‐24 Microwave 2 x 10‐24‐ 2 x 10‐22 Infrared 3 x 1011 ‐ 4 x 1014 Optical 4 x 10‐7 ‐ 7 x 10‐7 4 x 1014 ‐ 7.5 x 1014 UV 7.5 x 1014 ‐ 3 x 1016 5 x 10‐19 ‐ 2 x 10‐17 X‐ray 1 x 10‐11 ‐ 1 x 10‐8 Gamma‐ray < 1 x 10‐11 > 3 x 1019 > 2 x 10‐14 2. Stellar spectra Spectra can be made in many ways: with a prism (as Newton did in the 1660's), with drops of water (as in a real rainbow), or with gratings. Astronomers produce spectra by means of a "spectrograph" attached to the telescope. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Figure 3: Credit: Australia Telescope Outreach and
Education. Adapted from a diagram by James B. Kaler, in "Stars and their Spectra," Cambridge University Press, 1989.
In the modern spectrograph, light is sent from the telescope onto a narrow slit, whose mission is to limit the amount of light entering the spectrograph so that it acts as a point source of light from a larger image. Light is sent to a collimator, a curved mirror that makes the beam of rays parallel. The collimator sends the beam to a reflecting grating. A diffraction grating has thousands of narrow lines ruled onto a glass surface. The colored light is then focused by a camera onto a detector, usually a "charge‐coupled device" (CCD), that records the spectra digitally. The comparison lamp provides spectral lines of known wavelength, allowing the conversion from a pixel scale (that of the CCD) to a physical scale (e.g. angstroms). 2.1. Components of the stellar spectrum There are three components of a stellar spectrum: continuum emission (blackbody radiation), emission lines, and absorption lines. •
Continuum emission: This type of emission is caused by an opaque material which emits radiation because of its temperature. For example, the core of a star acts as a black body radiator. •
Absorption spectrum: An absorption line is characterized by a lack of radiation at specific wavelength. Absorption lines are created by viewing a hot opaque object through a cooler, thin gas. The cool gas in front absorbs some of the continuum emission from the background source, and re‐emits it in another direction, or at another frequency. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Absorption lines are subtracted from the continuum emission, so that they appear fainter. The "strength" of an absorption line ‐‐ the amount of energy removed from the spectrum ‐‐ depends on the amount of the particular chemical element in the star causing the line and on the efficiency of absorption. Figure 4: Different components of a stellar spectrum. Credit: Australia
Telescope Outreach and Education. Originally adapted from J.B. Kaler
"Stars and their spectra" Cambridge University Press, 1989 •
Emission spectrum: An emission line is characterized by excessive radiation at specific wavelengths. Emission lines are added to the continuum emission, so that they appear brighter. You can observe emission lines by looking at the cool gas (that caused absorption lines above) from the 'side', away from the background source. Some of the photons that were redirected in the production of absorption lines will come your way, but there will be only small amounts of continuum radiation. The important thing to know about absorption and emission lines is that every atom of a particular element (hydrogen, say) will have the same pattern of lines all the time. And the spacing of the lines is the same in both absorption and emission, only, emission lines are added to the continuum, while absorption lines are subtracted. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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2.2. Characterization of the spectral line profile The strength and width of a spectral line provide important information about the physical properties of the star. For example, the strength of certain lines correlates with the effective temperature of the star. The line width depends on the density and pressure of the region where it is produced. The line may appear shifted from its nominal central wavelength. This shift may provide a value of the distance to the star. The parameters most widely used to characterize the strength and width of a spectral line are the equivalent width (and peak intensity) and the Full Width at Half Maximum. Figure 5: Definition of equivalent width of a spectral line Equivalent Width (EW) is defined as the section of a surface counted between the level of the continuum, normalized to unity, and reference zero, having a surface identical to the profile of the line (see figure below). The equivalent width is thus measured in wavelength unit (in angstroms for example). The Peak intensity (P) is the ratio between the intensity at‐ the center of the line I and the intensity of the continuum Ic: Full Width at Half Maximum (FWHM) is the width measured at half level between the continuum and the peak of the line. The FWHM is expressed either in wavelength unit or in PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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speed unit when the objective is to measure expansion or disk speeds (if FWHM is in unit of wavelength, the width in km/sec is given by c . FWHM /λ, where c is the speed of the light = 3.105 km/s and λ is the central wavelength of the line). Figure 6: Definition of Full Width at Half
Maximum of a spectral line. In this case an
emission line. When the considered function is the Normal distribution, the FWHM is related to the standard deviation σ by the expression FWHM=2.35 σ. Notice that the FWHM measured from the star has to be corrected for instrumental width according to the equation: FWHMinstrument is the width that one measures on a very fine line. It is also the theoretical spectral resolution of the spectrograph. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Figure 7: Calibration lamp spectrum
The following exercises require the use of the IRAF tool splot. This is a very powerful tool that does many things. You should read the information given in the help facility in IRAF. For this project you basically must learn how to obtain equivalent widths and how to fit a Gaussian function to the line profile. Exercise 2: a) Estimate the spectral resolution of the spectrograph used by measuring the FWHM of a calibration line (see Figure), filename “nova_arc.fits”. b) Obtain the equivalent width of the lines in the spectrum “nova.fits”. c) Correct the FWHM for instrumental broadening. IRAF help. How to obtain the equivalent width. 1. open terminal xterm (or xgterm) and start IRAF (type cl) 2. move to the directory where you have the spectra 3. run splot nova_arc.fits xmin=6300 xmax=6850 (these numbers represent the lower and upper limits of the displayed wavelength range. Other lines require different values) 4. Place the mouse on the continuum at the left of the line making sure that you include the wings of the line and press e. Place the mouse on the continuum at the right of the line and press e again. IRAF will print out on the screen the value of the equivalent width. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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IRAF help. How to obtain the FWHM 5. open terminal xterm (or xgterm) and start IRAF (type cl) 6. move to the directory where you have the spectra 7. run splot nova_arc.fits xmin=4700 xmax=5100 (these numbers represent the lower and upper limits of the displayed wavelength range. Other lines require different values) 8. Normalize to unity: place the mouse on the IRAF terminal and press “t” followed by “/” and “q” 9. Place the mouse on the continuum at the left of the line making sure that you include ~20 amstrongs of continuum. Type “k”, place the mouse on the continuum at the right of the line and type “k” again. IRAF will perform a Gaussian fit to the line and the spectral parameters (center, flux, equivalent width and FWHM) will appear. 3. Stellar temperature Stars come in a wide range of sizes and temperatures. The hottest stars in the sky have temperatures in excess of 40,000 K, whereas the coolest stars that we can detect optically have temperatures on the order of 2,000‐3,000 K. The appearance of the spectrum of a star is very strongly dependent on its temperature. For example, the very hottest stars (called O‐
type stars) show absorption lines due to ionized helium (He II) and doubly or even triply ionized carbon, oxygen or silicon. On the other hand, the coolest stars (M‐type) show lines produced by molecules. Exercise 3: If all the stars in the Main Sequence are basically made up of hydrogen, why do we not see the same spectral lines in all the stars? The Differences among the spectral types are due to differences in Temperature. •
Which spectral lines you see depends primarily on the state of excitation and ionization of the gas. •
Excitation and Ionization are determined primarily by the temperature of the gas. •
Composition differences are unimportant. While the differences in spectra might seem to indicate different chemical compositions, in almost all instances, it actually reflects different surface temperatures. With some exceptions (e.g. the R, N, and S stellar types), there is no significant chemical or nuclear processing of the gaseous outer envelope of a star once it has formed. Fusion at the core of the star results in fundamental compositional changes, but material does not generally mix between the visible surface of the star and its core. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Hydrogen dominates the Sun, yet absorption lines of ionized calcium dominate the solar spectrum even though there is 440,000 times as much hydrogen as calcium. Hydrogen has a low efficiency of absorption, whereas that of ionized calcium is very high. The efficiency depends on the availability of electrons to move to higher energies and on atomic factors, namely the likelihood of absorption in the presence of a passing photon. The efficiencies depend critically on temperature and can be calculated from theory or measured in the laboratory. Once they are known, we can calculate the abundances of the atoms from the strengths of the absorption lines and therefore calculate the chemical composition of the outer part of a star. Relative absorption line strengths can also be used to find temperatures and densities. Absorption lines occur when an electron absorbs energy from the spectrum to move up the energy levels in the atom. Since hydrogen has only one electron, this electron is usually in the ground state. But as the temperature rises, the average electron gains more energy from collisions with other atoms, moving up to the second energy level, then the third, and so on. If the gas is hot enough, the electrons leave the atom entirely, so that it becomes ionized. There is a particular temperature at which the average electron will be in the second energy level. At this temperature, there are LOTS of atoms which can absorb Balmer line photons from the spectrum, and therefore stars of this temperature will have the strongest Balmer lines. In short, O stars do not show many lines because they have so high temperatures that atoms are ionized. K stars do no show lines because the temperature is too low to make the electrons to populate higher levels other than the ground state. So a lack of hydrogen absorption lines in a star does not necessarily mean the star's atmosphere is devoid of Hydrogen, it could also mean that the star has a low or very high surface temperature. In this project we shall estimate the temperature of a star following two different procedures: i) Wien's law and ii) the strength (i.e., the equivalent width) of some spectral lines (Hα and Na). 3.1. Estimating the effective temperature of a star from the Wien's law Wien's displacement law is a law of physics that states that there is an inverse relationship between the wavelength of the peak of the emission of a blackbody and its temperature. λmax = b/T where λmax is the peak wavelength in meters T is the temperature of the blackbody in kelvins (K), and b is a constant of proportionality, called Wien's displacement constant and equals 2.897768 5(51) × 10–3 m K. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Note that the spectrum lets us determine the surface temperature (not the core temperature) of the star because the radiation we measure as the star's spectrum comes from the surface layers of the star. This is called the effective temperature of the star and corresponds to the temperature of a black body having the same size and luminosity as the star. Figure 8: Blackbody curves for thermal bodies at different
temperatures. Note how the peak flux shifts toward larger
wavelength as the temperature increases. Look at Figure 8. Hotter objects emit most of their radiation at shorter wavelengths; hence they will appear to be bluer. Cooler objects emit most of their radiation at longer wavelengths; hence they will appear to be redder. Furthermore, at any wavelength, a hotter object radiates more (is more luminous) than a cooler one. The total energy radiated is proportional to the area under the graph. Figure 9: Spectra of star of different temperature. The blue line indicates the blackbody curve. Credit: Prof. Rowan's web page. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Exercise 4: While the absorption lines are determined by the presence or absence of particular ions at different temperatures, the shape of the continuum is determined by the blackbody radiation laws. In either case, the temperature is the main parameter driving the differences between spectra. Look at the three spectra shown in the Figure below. Can you tell from the continuum which star is hottest? and the coolest? Figure 10: Spectra of stars with different effective temperature
taken from the Skinakas Observatory. Arrows mark the peak of the
emission. Exercise 5: Following the example given in Fig. 11, estimate the effective temperature of the two spectra given in Fig.12. Figure 11: Application of Wien's law to estimate the effective temperature
of a star. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Figure 12: Estimate the effective temperature of these two spectra. 3.2. Estimating the effective temperature of a star from the spectral lines Instead of measuring the entire spectrum of a star, we can estimate its temperature by looking at only certain wavelengths of the spectrum. One of the best examples of this temperature sensitivity comes by comparing the Balmer lines, which are lines produced by neutral hydrogen atoms. The strength of these lines in the spectrum of a star is an excellent indication of the temperature of that star. However, as we saw above, early‐type (O and B) and late‐type (K and M) have very weak Balmer lines. A‐type stars have very strong hydrogen lines. Thus, the strength of the Hα will increase from O to A stars and decrease after. This means that very early‐type and very late‐type stars may have similar values of Hα equivalent width (EW(Hα)). We need another indicator of the temperature so that we break the degeneracy. The sodium Na doublet at 5890‐5896 Å has a well‐known behaviour related to the spectral type and luminosity class. In order to apply this method one needs to know the relationship between the strength of the Hα and Na lines and the temperature. Such relationship is called a calibration. As a measure of the strength of the lines we shall use the equivalent width. The Table below shows a list of MK standard stars. The values of the effective temperature were taken from Theodosiou & Danezis (1991, Ap&SS,183,91) for stars with spectral types B‐F and Bell & Gustaffson (1989, MNRAS, 236, 653) for G‐K type stars. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Star Spectral type EW(Hα) (Å) EW(Na) (Å) Temperature (k) HD 886 B2IV 22400 HD120315 B3V 18445 HR 9087 B7IV 12915 HR 8634 B8V 12120 HD196867 B9IV 11020 HR 5501 B9.5V 10340 HD130109 A0III 10240 HR 262 A3V 8625 HD116842 A5V 8170 HD 4758 A9III 7500 HD 223486 F0V 7500 HD 6680 F3IV 6500 HD134083 F5V 6500 HD 6301 F7IV 6000 HD196755 G2IV 5650 HD153751 G5III 5600 XiHer G8III 5000 HD 217143 G9III 4850 HD145675 K0V 5300 HD198550 K4V 4750 Exercise 6: Measure the equivalent width of the Hα and Na lines from the spectra of the stars given in the Table and complete the Table. If your measurements were correct you should observe that the equivalent width of the Hα, EW(Hα), and Na, EW(Na), lines correlate with the effective temperature. It is monotonic in the case of the Na line, that is, as the equivalent width increases the temperature decreases and non‐monotonic for the Hα line. For the Hα line the equivalent width first increases as the temperature increases and then decreases. Also, since the EW(Na) is very small or even this line may be absent in very early‐type stars it is not a very convenient proxy to derive the effective temperature. PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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In order to estimate the temperature is a star we shall proceed as follows: we first separate the stars we want to measure into “hot” and “cold” stars according to the equivalent width of the Na lines. For each one of this groups the temperature is a monotonic function of the EW(Hα). The boundary between hot and cold stars is given by spectral type A0, as it is for this spectral type that EW(Hα) is maximum. Then we can use EW(Hα) to determine the temperature more precisely. So, if the EW(Na) of the unknown object is smaller than that of the A0 star then the object falls in the “hot” category. If it is larger it will be classified as a “cold” star. For each one of these two groups, hot and cold, fit a function with EW(Hα) as the independent variable and Teff as the dependent variable. The resulting equation will be the EW‐Teff calibration. Exercise 7: Create a plot (see Fig. 13) of the temperature as a function of the Hα equivalent width separating into hot and cold stars and fit a function (cold stars can be fitted with a linear function or straight line; hot stars can also be fitted with a power‐law function). Figure 13: Effective temperature as a function of the strength of the Halpha and Na lines PROJECT 9: STELLAR SPECTRA: TEMPERATURE
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Exercise 8: Measure the equivalent width of the stars given in the Table below and estimate their effective temperature and spectral type. The Table below gives a list of stars for which the temperature is unknown. Calculate the EW(Na) and classified the stars as “hot” or “cold”. Then calculate EW(Hα) and estimate the temperature by applying the calibration obtained above. Name Type (early/late) EW(Hα) (Å) Temperature (K) Spectral type HR718 HD 137391 HD 160269 PROJECT 9: STELLAR SPECTRA: TEMPERATURE