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E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 1 of 29 pages 5th class Feb. 25, 2013 Outline for PHS 207 – The Stars – Spectra and Luminosity 1. Quiz on the Sun Label the 6 layers of the Sun and the 2 features Picture from http://t0.gstatic.com/images?q=tbn:ANd9GcRHKEB7P1935hoRjAGBYSjgbh8NYRZOXxTFQycAjiiudvHMglI4_w Part 1 2 3 4 5 6 Corona Temperature (oK in Kelvin degrees) 2,000,000 Description/Function/Result Outermost; seen only during solar eclipse E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 2 of 29 pages 2. Introduction to NAAP ClassAction Questions #4 Number line – EM Spectrum Spectra or spectral lines indicate what elements are present in a star and its overall color its temperature. 3 The light seen from a star is typically an absorption spectrum. The light produced by the surface of the star – the photosphere – is a continuous spectrum meaning that all wavelengths of light are present. However, certain wavelengths of light are redirected (absorbed and re-emitted in random directions) in the cooler low-density layers above the surface (the chromosphere) of a star. This occurs because electrons in Hydrogen (and other atoms) absorb light as they jump to higher orbitals and then re-emit the light as they drop back down. Thus, we see absorption spectra from stars – rainbows with dark gaps known as spectral lines – because a lot of the light is missing at particular wavelengths 4. . Hydrogen Energy Level Transitions Absorbing Photons Because an electron bound to an atom can only have certain energies the electron can only absorb photons of certain energies. According to the theory quantum mechanics, an electron bound to an atom can not have any value of energy, rather it can only occupy certain states which correspond to certain energy levels. The formula defining the energy levels of a Hydrogen atom are given by the equation: E = -E0/n2, where E0 = 13.6 eV (1 eV = 1.602×10-19 Joules) and n = 1,2,3… and so on. The energy is expressed as a negative number because it takes that much energy to unbind (ionize) the electron from the nucleus. It is common convention to say an unbound E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 3 of 29 pages electron has zero (binding) energy. Because an electron bound to an atom can only have certain energies the electron can only absorb photons of certain energies exactly matched to the energy difference, or “quantum leap”, between two energy states. When an electron absorbs a photon it gains the energy of the photon. Because an electron bound to an atom can only have certain energies the electron can only absorb photons of certain energies. For example an electron in the ground state has an energy of -13.6 eV. The second energy level is -3.4 eV. Thus it would take E2 − E1 = -3.4 eV − -13.6 eV = 10.2 eV to excite the electron from the ground state to the first excited state. If a photon has more energy than the binding energy of the electron then the photon will free the electron from the atom – ionizing it. The ground state is the most bound state and therefore takes the most energy to ionize. Emitting Photons When an electron drops from a higher level to a lower level it sheds the excess energy, a positive amount, by emitting a photon. Generally speaking, the excited state is not the most stable state of an atom. An electron has a certain probability to spontaneously drop from one excited state to a lower (i.e. more negative) energy level. When an electron drops from a higher level to a lower level it sheds the excess energy, a positive amount, by emitting a photon. The energy of the emitted photon is given by the Rydberg Formula. This formula is essentially the subtraction of two energy levels. It is: – Rydberg Formula 1 1 Ephoton = E0 n 2 − n 2 1 2 ( ) where n1 < n2 and (as before) E0 = 13.6 eV. With the restriction n1 < n2 the energy of the photon is always positive. This means that the photon is emitted and that interpretation was the original application of Rydberg. It also works if the n1, n2 restriction is relaxed. In that case the negative energy means a photon (of positive energy) is absorbed. For Hydrogen some of the (emitting photon transitions) are named for Lyman, Balmer, and Paschen (Lines) those who discovered them. E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 4 of 29 pages Long before the Hydrogen atom was understood in terms of energy levels and transitions, astronomers had being observing the photons that are emitted by Hydrogen (because stars are mostly Hydrogen). Atomic physicist Balmer noted, empirically, a numerical relationship in the energies of photons emitted. This relationship was generalized and given context by the Rydberg Formula. But the various discrete photon energies/wavelengths that were observed by Balmer were named the Balmer series. It was later understood that the Balmer lines are created by energy transitions in the Hydrogen atom. Specifically, when a photon drops from an excited state to the second orbital, a Balmer line is observed. The Balmer series is important because the photons emitted by this transition are in the visible regime. The Balmer series is indicated by an H with a subscript α, β, γ, etc. with longest wavelength given by α. As there are other transitions possible, there are other “series”. All transitions which drop to the first orbital (i.e. the ground state) emit photons in the Lyman series. All transitions which drop to the 3rd orbital are known as the Paschen series. The graphic to the right shows some of the Lyman and Balmer transitions graphically. Spectra Lines Lyman Balmer Transitions to Ground state 1st Excited state E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 5 of 29 pages Paschen 2nd Excited state Hydrogen Atom Simulator http://astro.unl.edu/classaction/animations/light/hydrogenatom.html Hydrogen Atom Simulator – Introduction The Hydrogen Atom Simulator allows one to view the interaction of an idealized Hydrogen atom with photons of various wavelengths. This atom is far from the influence of neighboring atoms and is not moving. The simulator consists of four panels. Below gives a brief overview of the basics of the simulator. The panel in the upper left shows the Bohr Model: the proton, electron, and the first six orbitals with the correct relative spacing. o The electron can absorb photons and jump higher energy levels where it will remain for a short time before emitting a photon(s) and drop to lower energy level (with known probabilities fixed by quantum mechanics). o The electron can also be ionized. The simulator will a short time later absorb an electron. E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 6 of 29 pages o For convenience you can drag the electron between levels. Once it is released it will behave “physically” once again as if it had gotten to that present level without being dragged. The upper right panel labeled “energy level diagram” shows the energy levels vertically with correct relative spacing. The “Photon Selection” panel (bottom left) allows one to “shoot” photons at the Hydrogen atom. The slider allows the user to pick a photon of a particular energy/wavelength/frequency. o Note how energy and frequency are directly proportional and energy and wavelength are inversely proportional. o On the slider are some of the energies which correspond to levels in the Lyman, Balmer, and Paschen series. o Clicking on the label will shoot a photon of that energy. o If the photon is in visual band, its true color is shown. Photons of longer wavelengths are shown as red and shorter wavelengths as violet. The “Event Log” in the lower right lists all the photons that the atom has encountered as well as all the electron transitions. o The log can be cleared by either using the button or manually dragging the electron to a particular energy level. Hydrogen Atom Simulator – Exercises For any particular level of the Hydrogen atom one can think of the photons that interact with it as being in three groups: Range 1 Increasing Energy → Range 2 Range 3 All the photons have enough energy to ionize the atom. Some of the photons have the right energy to make the electrons to jump to a higher energy level (i.e. excite them). Note that the ranges are different for each energy level. Below is an example of the ranges for an electron in the ground state of a Hydrogen atom. None of the photons have enough energy to affect the atom. Range 1 Ground State electron of H Range 2 Range 3 0eV to 10.2 eV 10.2 to 13.6 >13.6 eV (10.2 eV needed to excite electron to 1st orbital) (some will excite, some won’t) (anything greater than this will ionize the electron) E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 7 of 29 pages When the simulator first loads, the electron is in the ground state and the slider is at 271 nm. Fire a 271 nm photon. This photon is in range 1. Gradually increase the slider to find a photon which is between range 1 and range 2 (for a ground state electron). This should be the Lyman-α line (which is the energy difference between the ground state and the second orbital). Increase the energy a bit more from the Lyman-α line and click “fire photon”. Note that nothing happens. This is a range 2 photon but it doesn’t have the “right energy”. Increase the energy more until photons of range 3 are reached. In the simulator this will be just above the Lε line. o Technically there are photons which would excite to the 7th, 8th, 9th, etc. energy levels, but these are very close together and those lines not shown on the simulator. o The Lε line has an energy of -13.22 eV and is in range two. The ionization energy for an electron in the ground state is 13.6 eV and so that is the correct range 3 boundary. Question 4: Which photon energies will excite the Hydrogen atom when its electron is in the ground state? (Hint: there are 5 named on the simulator, though there are more.) Question 5: Starting from the ground state, press the Lα button twice in succession (that is, press it a second time before the electron decays). What happens to the electron? Question 6: Complete the energy range values for the 1st excited state (i.e. the second orbital) of Hydrogen. Use the simulator to fill out ranges 2 and range 3. The electron can be placed in the 1st orbital by manually dragging the electron or firing a Lα photon once when the electron is in the ground state. Note also that the electron will deexcite with time and so it may need to be placed in the 1st orbital repeatedly. Range 1 0 to 1.9 eV (anything less than this energy will fail to excite the atom) 1st Excited State Electron in H Range 2 Range 3 E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 8 of 29 pages Question 7: What is the necessary condition for Balmer Line photons (H α , etc) to be absorbed by the Hydrogen atom? _____________________________________________ Question 8: Complete the energy range values for the 3 rd orbital (2nd excited state) of Hydrogen. The electron can be placed in the 3rd orbital by manually dragging the electron or firing an Lβ photon once when the electron is in the ground state. Note also that the electron will deexcite with time and so it may need to be placed in the 2nd orbital repeatedly. 56 56 4 4 3 3 Hδ 2 2 Lα 1 1 E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 9 of 29 pages Question 8: Complete the energy range values for the 3 rd orbital (2nd excited state) of Hydrogen. The electron can be placed in the 3rd orbital by manually dragging the electron or firing an Lβ photon once when the electron is in the ground state. Note also that the electron will deexcite with time and so it may need to be placed in the 2nd orbital repeatedly. Range 1 3rd Electron Orbital in H Range 2 Range 3 >1.5 eV (anything more than this will ionize the atom) Question 9: Starting from the ground state, press two and only two buttons to achieve the 6th orbital in two different ways. One of the ways has been given. Illustrate your transitions with arrows on the energy level diagrams provided and label the arrow with the button pressed. Metthod 1 Method 2 56 56 4 4 3 3 Hδ 2 2 b) Method 2: Lα 1 1 E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 10 of 29 pages Question 10: Press three buttons to bring the electron from the ground state to the 4th orbital. Illustrate the transitions as 56 4 arrows on the energy level diagrams provided and label the 3 arrow with the button pressed. 2 1 Question 11: How does the energy of a photon emitted when the electron moves from the 3rd orbital to the 2nd orbital compare to the energy of a photon absorbed when the electron moves from the 2nd orbital to the 3rd orbital? ____________________________________ Question 12: Compare the amount of energy needed for the following 3 transitions. Explain why these values occur. Lα: Level 1 to Level 2 _______________ Hα: Level 2 to Level 3 _______________ Pα: Level 3 to Level 4 E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 11 of 29 pages Black Body Radiation Curve is produced by temperature (Thermal Distribution) of the atoms of the star. 5. Line Strength is the intensity of the absorption or emissions. When Hydrogen is excited it emits light as photons de-excite. Or conversely, the Hydrogen will absorb photons of certain energies. The strength of the line from a source of Hydrogen will depend on how many electrons are in a particular excited state. If only very few electrons are the first excited state, the Balmer lines will be very weak. If many Hydrogen atoms are in the first excited state then the Balmer lines will be strong. The number of Hydrogen atoms are in what state is a statistical distribution that depends on the temperature of the Hydrogen source. The Thermal Distribution simulator shown later demonstrates this. Use the “Thermal Distribution Simulator” below this graph to plot of the number of atoms with electrons in the 2nd orbital. There should be at least 8 points on your plot. Note also that the y-axis is in terms of 1015 particles. Thus the point for 15,000 K, which has 3.82 × 1017 particles in the 2nd orbital, will read as 382 on the graph. Plot at least 8 points on the graph. More points near “interesting” features is highly recommended. Fit (draw) a curve to the plotted points. It should be a smooth curve. E:\2012-2013\SSU\PHS 207 spring 2013\Notes …\Stars\Document1 12 of 29 pages The light seen from a star typically contains an absorption spectrum. The light produced by the surface of the star – the photosphere – is a continuous spectrum meaning that all wavelengths of light are present. However, certain wavelengths of light are redirected (absorbed and reemitted in random directions) in the cooler low-density layers above the surface (the chromosphere) of a star. This occurs because electrons in Hydrogen (and other atoms) absorb light as they jump to higher orbitals and then re-emit the light as they drop back down. Thus, we see absorption spectra from stars – rainbows with dark gaps known as spectral lines – 3500 number of levl 2 atoms (1E15) 3000 2500 2000 1500 1000 500 0 3000 6000 9000 12000 15000 18000 21000 24000 27000 30000 Temperature (K) because a lot of the light is missing at particular wavelengths. E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 13 of 29 pages Question 13: Consider the strength of the Hβ absorption line in the spectra of stars of various surface temperatures. This is the amount of light that is missing from the spectra because Hydrogen electrons have absorbed the photons and jumped from level 2 to level 4. How do you think the strength of Hβ absorption varies with stellar surface temperature? A way to measure the average speed of the atoms with respect to each other is temperature. Thermal Distribution of Hydrogen atoms From http://astro.unl.edu/naap/hydrogen/abundances.html E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 14 of 29 pages One Atom vs. Many Atoms When an atom is by itself, in isolation, its orbitals behave differently than when in packed tightly with other atoms. For example, the energy levels of a single carbon atom are slightly different than a diamond. Similarly, hydrogen gas (H2) is a tiny bit different than a simple H atom. The difference in energy levels, however, is not much in that case. The Hydrogen Atom Simulator showed just one H atom. Astronomically one H atom is never observed. Rather, only vast numbers of Hydrogen atoms together are observed. Often they atoms (or H2 molecules) are on average far enough apart so that the orbitals aren't significantly altered. Another way of saying that is the density is “low”. But even when the density is low (which we will assume here), there are the occasional collisions between the atoms. When they collide some of the energy goes into them bouncing off of each other and some of it can go into exciting electrons. How frequently collisions occur and how much energy typically goes to exciting the atom depends on how fast the low density cloud of hydrogen atoms are moving on average. A way to measure the average speed of the atoms with respect to each other is temperature. Thermal Distribution The histogram above is of 1025 Hydrogen atoms. The density of the atoms is low so that their energy levels are very close to what we see for a single atom. The temperature can be varied from 3000 K to 30,000 K (a range which includes the surface temperature of almost all stars). Experiment with the slider the histogram to change the temperature for the 8 data points for the graph on the previous page. The histogram can be downloaded from http://astro.unl.edu/naap/hydrogen/abundances.html. Note that at 3000 K almost all of the atoms are in the ground state. Note also that at 3000 K there are more ionized atoms than there are atoms with electrons in the 2nd orbital. As the temperature is increased the disparity between the number of ionized atoms and level 2 atoms will become even larger. This occurs because of the energy spacing of the levels – an electron in level 1 is much more tightly bound than one in level 2. Collisions between atoms that are sufficiently energetic to knock the electron from the ground state to level 2, but not sufficiently energetic to ionize that atom become rare as the temperature rises. Note also that the histogram is logarithmic and the relative heights of the bars behave in a non-intuitive fashion. For example, at 20,700 K the ionized atom bar is only about 20% higher than the level 1 bar but there are 10,000 ionized atom to every level 1 atom. http://astro.unl.edu/naap/blackbody/animations/blackbody.html E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 15 of 29 pages Filters Simulator Overview The filters simulator allows one to observe light from various sources passing through multiple filters and the resulting light that passes through to some detector. An “optical bench” shows the source, slots for filters, and the detected light. The wavelengths of light involved range from 380 nm to 825 nm which more than encompass the range of wavelengths detected by the human eye. The upper half of the simulator graphically displays the source-filter-detector process. A graph of intensity versus wavelength for the source is shown in the leftmost graph. The middle graph displays the combined filter transmittance – the percentage of light the filters allow to pass for each wavelength. The rightmost graph displays a graph of intensity versus wavelength for the light that actually gets through the filter and could travel on to some detector such as your eye or a CCD. Color swatches at the far left and right demonstrate the effective color of the source and detector profile respectively. The lower portion of the simulator contains tools for controlling both the light source and the filter transmittance. In the source panel perform the following actions to gain familiarity. o Create an blackbody source distribution – the spectrum produced by a light bulb which is a continuous spectrum. Practice using the temperature and peak height controls to control the source spectrum. o Create a bell-shaped spectrum. This distribution is symmetric about a peak wavelength. Practice using the peak wavelength, spread, and peak height controls to vary the source spectrum. o Practice creating piecewise linear sources. In this mode the user has complete control over the shape of the spectrum as control points can be dragged to any value of intensity. Additional control points are created whenever a piecewise segment is clicked at that location. E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 16 of 29 pages Control points may be deleted by holding down the Delete key and clicking them. Control points can be dragged to any location as long as they don’t pass the wavelength value of another control point. In the filters panel perform the following actions to gain familiarity. o Review the shapes of the preset filters (the B,V, and R filters) in the filters list. Clicking on them selects them and displays them in the graph in the filters panel. o Click the add button below the filters list. Rename the filter from the default (“filter 4”). Shape the piecewise linear function to something other than a flat line. o Click the add button below the filters list. Select bell-shaped from the distribution type pull down menu. Alter the features of the default and rename the filter. o If desired, click the remove button below the filters list. This removes the actively selected filter (can’t remove the preset B,V, and R filters). Filters are not saved anywhere. Refreshing the flash file deletes the filters. Filters Simulator Questions Use the piecewise linear mode of the source panel to create a “flat white light” source at maximum intensity. This source will have all wavelengths with equal intensity. Drag the V filter to a slot in the beam path (i.e. place them in the filter rack). Try the B and the R filter one at a time as well. Dragging a filter anywhere away from the filter rack will remove it from the beam path. Question 1: Sketch the graphs for the flat white light and V filter in the boxes below. What is the effective color of the detected distribution? __________________________________________ source distribution combined filter transmittance detected distribution Question 2: With the flat white light source, what is the relationship between the filter transmittance and the detected distribution? ______________Add a new piecewise linear filter. E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 17 of 29 pages Adjust the filter so that only large amounts of green light pass. This will require that addition of points. Question 3: Use this green filter with the flat white light source and sketch the graphs below. source distribution combined filter transmittance detected distribution Question 4: Use the blackbody option in the source panel to create a blackbody spectrum that mimics white light. What is the temperature of this blackbody you created? ________________ • Add a new piecewise linear filter to the filter list. Modify the new filter to create a 40% “neutral density filter”. That is, create a filter which allows approximately 40% of the light to pass through at all wavelengths. Set up the simulator so that light from the “blackbody white light” source passes through this filter. Question 5: Sketch the graphs created above in the boxes below. (This situation crudely approximates what sunglasses do on a bright summer day.) source distribution combined filter transmittance detected distribution Question 6: Remove all filters in the filters rack. Place a B filter in the beam path with the flat white light source. Then add a second B filter and then a third. Describe and explain what happens when you add more than one of a specific filter. E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 18 of 29 pages Question 7: Place a B filter in the beam path with the 40% neutral density filter. Then add a V filter into the beam path. Describe and explain what happens when you add more than one filter to the filter rack. purple filter Question 8: Create a piecewise linear filter that when used with the flat white light source would allow red and blue wavelengths to pass and thus effectively allowing purple light to pass. Draw the filter in the box to the right. • Remove all filters from the filters rack. Create a very narrow bell-shaped source distribution that is peaked at green wavelengths (somewhere close to 550 nm). Notice the color. Expand the spread of the source distribution to maximum. Notice how the color changes. Change the distribution source to an blackbody source peaked at green wavelengths (a temperature close to 5270 K). Again notice the color. Question 9: Using observations from the above actions, explain why we don’t observe “green stars” in nature, though there are indeed stars which emit more green light than other wavelengths. ___________________________________________________________________ 6. Absolute and Apparent Brightness Magnitude Apparent Magnitude From http://www.phys.ksu.edu/personal/wysin/astro/magnitudes.html Apparent magnitude m of a star is a number that tells how bright that star appears at its great distance from Earth. The scale is "backwards" and logarithmic. Larger magnitudes correspond to fainter stars. Note that brightness is another way to say the flux of light, in Watts per square meter, coming towards us. On this magnitude scale, a brightness ratio of 100 is set to correspond exactly to a magnitude difference of 5. As magnitude is a logarithmic scale, one can always transform a brightness ratio B2/B1 into the equivalent magnitude difference m2-m1 by the formula: E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 19 of 29 pages m2-m1 = -2.50 log(B2/B1). You can check that for brightness ratio B2/B1=100, we have log(B2/B1) =log(100)= log(102) = 2, and then m2-m1=-5, the basic definition of this scale (brighter is more negative m). One then has the following magnitudes and their corresponding relative brightnesses: magnitude m | 0 1 2 3 4 5 6 7 8 9 10 ---------------------------------------------------------------relative | 1 2.5 6.3 16 40 100 250 630 1600 4000 10,000 brightness | ratios | (Note that the lower row of numbers is just (2.512)m.) Absolute Magnitude Absolute magnitude Mv is the apparent magnitude the star would have if it were placed at a distance of 10 parsecs from the Earth. Doing this to a star (it is a little difficult), will either make it appear brighter or fainter. From the inverse square law for light, the ratio of its brightness at 10 pc to its brightness at its known distance d (in parsecs) is B10/Bd=(d/10)2. Then, like the formula above, we say that its absolute magnitude is Mv = m - 2.5 log[ (d/10)2 ]. Stars farther than 10 pc have Mv more negative than m, that is why there is a minus sign in the formula. If you use this formula, make sure you put the star's distance d in parsecs (1 pc = 3.26 ly = 206265 AU). Distance Determination The above relation can also be used to determine the distance to a star if you know both its apparent magnitude and absolute magnitude. This would be the case, for example, when one uses Cepheid or other variable stars for distance determination. Turning the formula inside out: d = (10 pc) x 10(m-Mv)/5 For example, for a Cepheid variable with Mv = -4, and m = 18, the distance is E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 20 of 29 pages d = (10 pc) x 10[18-(-4)]/5 = 2.51 x 105 pc. 7. Measuring Stellar Distance The Cosmic Distance Ladder There are at least seven different astronomical distance determination techniques: -- radar ranging, -- parallax, -- distance modulus, -- spectroscopic parallax, -- main sequence fitting, -- Cepheid variables, and --supernovae On Monday February 25, 2013 we will only look at parallax while distance modulus and spectroscopic parallax are also given below. Parallax In addition to astronomical applications, parallax is used for measuring distances in many other disciplines such as surveying. http://astro.unl.edu/naap/distance/animations/parallaxExplorer.html Open the Parallax Explorer where techniques very similar to those used by surveyors are applied to the problem of finding the distance to a boat out in the middle of a large lake by finding its position on a small scale drawing of the real world. The simulator consists of a map providing a scaled overhead view of the lake and a road along the bottom edge where our surveyor represented by a red X may be located. The surveyor is equipped with a theodolite (a combination of a small telescope and a large protractor so that the angle of the telescope orientation can be precisely measured) mounted on a tripod that can be moved along the road to E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 21 of 29 pages establish a baseline. An Observer’s View panel shows the appearance of the boat relative to trees on the far shore through the theodolite. map. is a Configure the simulator to preset A which allows us to see the location of the boat on the (This helpful simplification to help us get started with this technique – normally the main goal of the process is to learn the position of the boat on the scaled map.) Drag the position of the surveyor around and note how the apparent position of the boat relative to background objects changes. Position the surveyor to the far left of the road and click take measurement which causes the surveyor to sight the boat through the theodolite and measure the angle between the line of sight to the boat and the road. Now position the surveyor to the far right of the road and click take measurement again. The distance between these two positions defines the baseline of our observations and the intersection of the two red lines of sight indicates the position of the boat. We now need to make a measurement on our scaled map and convert it back to a distance in the real world. Check show ruler and use this ruler to measure the distance from the baseline to the boat in an arbitrary unit. Then use the map scale factor to calculate the perpendicular distance from the baseline to the boat. Question 2: Enter your perpendicular distance to the boat in map units. ______________ Show your calculation of the distance to the boat in meters in the box below. Configure the simulator to preset B. The parallax explorer now assumes that our surveyor can make angular observations with a typical error of 3°. Due to this error we will now describe an area where the boat must be located as the overlap of two cones as opposed to a definite location that was the intersection of two lines. This preset is more realistic in that it does not illustrate the position of the boat on the map. Question 3: Repeat the process of applying triangulation to determine the distance to the boat and then answer the following: What is your best estimate for the perpendicular distance to the boat? What is the greatest distance to the boat that is still observations? consistent with your E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 22 of 29 pages What is the smallest distance to the boat that is still observations? consistent with your Configure the simulator to preset C which limits the size of the baseline and has an error of 5° in each angular measurement. Question 4: Repeat the process of applying triangulation to determine the distance to the boat and then explain how accurately you can determine this distance and the factors contributing to that accuracy. ________________________________________________ Distance Modulus Question 5: Complete the following table concerning the distance modulus for several objects. Object Star A Apparent Magnitude Absolute Magnitude m 2.4 M Star B 10 Star D 8.5 Distance (pc) 10 5 Star C Distance Modulus m-M 16 25 0.5 Question 6: Could one of the stars listed in the table above be an RR Lyrae star? Explain why or why not. __________________________________________________________ Spectroscopic Parallax E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 23 of 29 pages Open up the Spectroscopic Parallax Simulator. There is a panel in the upper left entitled Absorption Line Intensities – this is where we can use information on the types of lines in a star’s spectrum to determine its spectral type. There is a panel in the lower right entitled Star Attributes where one can enter the luminosity class based upon information on the thickness of line in a star’s spectrum. This is enough information to position the star on the HR Diagram in the upper right and read off its absolute magnitude. http://astro.unl.edu/naap/distance/animations/spectroParallax.html Let’s work through an example. Imagine that an astronomer observes a star to have an apparent magnitude of 4.2 and collects a spectrum that has very strong helium and moderately strong ionized helium lines – all very thick. Find the distance to the star using spectroscopic parallax. Let’s work through an example. Imagine that an astronomer observes a star to have an apparent magnitude of 4.2 and collects a spectrum that has very strong helium and moderately strong ionized helium lines – all very thick. Find the distance to the star using spectroscopic parallax. E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 24 of 29 pages Let’s first find the spectral type. We can see in the Absorption Line Intensities panel that for the star to have any helium lines it must be a very hot blue star. By dragging the vertical cursor we can see that for the star to have very strong helium and moderate ionized helium lines it must either be O6 or O7. Since the spectral lines are all very thick, we can assume that it is a main sequence star. Setting the star to luminosity class V in the Star Attributes panel then determines its position on the HR Diagram and identifies its absolute magnitude as -4.1. We can complete the distance modulus calculation by setting the apparent magnitude slider to 4.2 in the Star Attributes panel. The distance modulus is 8.3 corresponding to a distance of 449 pc. Students should keep in mind that spectroscopic parallax is not a particularly precise technique even for professional astronomers. In reality, the luminosity classes are much wider than they are shown in this simulation and distances determined by this technique are probably have uncertainties of about 20%. Question 7: Complete the table below by applying the technique of spectroscopic parallax. Observational Data Analysis m Description of spectral lines Description of line thickness 6.2 strong hydrogen lines moderate helium lines very thin 13.1 strong molecular lines very thick 7.2 strong ionized metal lines moderate hydrogen lines very thick Cepheid Variables http://astro.unl.edu/naap/distance/cepheids.html M m-M d (pc) E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 25 of 29 pages Cepheid variables are pulsating variable stars similar to the RR Lyraes mentioned earlier on the distance modulus page. However, Cepheids have longer pulsation periods and they are larger stars. Cepheids have been extremely important as distance indicators for many years. Although they don’t all have the same average absolute magnitude as RR Lyraes do, they are more useful since they are brighter stars and can be observed at greater distances. Henrietta Leavitt in 1912 was the first to recognize that there was a relationship between the pulsation periods and the luminosities of Cepheids. She recognized that larger, brighter Cepheids have longer pulsation periods, although she was unaware of the exact relationship. Harlow Shapley later calibrated the Cepheids – relating the periods of pulsation to the absolute magnitudes which led to the first estimate of the size of the Milky Way. The calibration of the period-luminosity relationship has improved over time and a modern version is depicted in the figure to the right. E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 26 of 29 pages E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 27 of 29 pages To the left is a graph of the periodicity of the Type I (high-metallicity) Cepheid variable S Nor. Cepheid S Nor has a period of pulsation of approximately 10 days and an average apparent magnitude mV = 6.5, what is its distance? We can use the pulsation period to estimate the absolute magnitude of the Cepheid. From the chart above a period of 10 days corresponds to an absolute magnitude of -4. Thus, the distance modulus is m - M = 6.5 - (-4) = 10.5, which corresponds to a distance of 1260 pc. Note that our estimate is not particularly accurate since we didn’t take into account many subtleties that research astronomers would consider. Because of the brightnesses of Cepheids, astronomers can identify them in nearby galaxies. Observations of Cepheids by the Hubble Space Telescope have recently been used to estimate the distance to the Virgo Cluster which is about 18 Mpc E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 28 of 29 pages http://astro.unl.edu/naap/distance/summation.html The Cosmic Distance Ladder E:\2012-2013\SSU\PHS 207 spring 2013\Notes…\Stars\Document1 --- 29 of 29 pages The progression of distance determination techniques surveyed is collectively known as the Cosmic Distance Ladder. No one technique is effective at all distances and typically techniques useful at small distances are used to calibrate those used on objects at greater distances. Thus, one rung on the ladder allows astronomers to step to the next rung. As a first example of this, think about RR Lyrae stars. A number of RR Lyraes have been observed by the Hipparcos satellite which has determined accurate distances to them. It is easy to observe the apparent magnitudes of these stars and when they are combined with the known distances, the absolute magnitudes can be determined. This procedure allowed astronomers to see that all RR Lyrae have absolute magnitudes around 0.5. Astronomers can now calculate the distance to RR Lyraes beyond Hipparcos' range using the distance modulus. When astronomers find RR Lyraes in the Large Magallenic Cloud, they can be used to calibrate the PeriodLuminosity relation for Cepheids observed there. Similarly, when Type I supernovae in nearby galaxies with Cepheids in them (and thus known distances) were observed, the absolute magnitude of the supernova peak brightness was determined. Thus, a variety of distance determination techniques allow us to – one step at a time – learn the distances to ever more distant objects. THE END