text: Swihart: Quantitative Astronomy --class1-- Read: ch.1 in Swihart you need to have a firm understanding of "sky vocabulary" The following (from non-major astro) are tutorials & quizzes that force you to exercise that vocabulary Go to web: http://www.physics.csbsju.edu/astro/ Each of the following links opens to a sequence of multiple pages ending with an on-line quiz. Read the material; take the quizzes # tutorial: sky vocabulary # tutorial & quiz: sky map # tutorial & quiz: star finder # tutorial & quiz: SC001 star map # tutorial & quiz: SC002 star map # tutorial: latitude & your sky The sky is mapped in terms of "right ascension" & "declination" note that RA has the unusual unit of hours, where 24h=360deg, and both are commonly expressed in sexagesimal (base 60: hms, deg'") Internally (and increasingly externally) locations must be encoded like any real number; converting between the two systems takes an extra step. My calculator actually has a button "h.mmss" that does this conversion: if you want to convert 11h 3m 43.67s (or 11 deg 3' 43.67") you enter 11.034367, push the button and the calculator gives: 11.0621306=11+3/60+43.67/3600 Point: If you are working problems on your calculator finding this button can save you time. Most online astro apps understand sexagesimal but the exact data entry format can vary, e.g., 11:03:43.67 or 11 03 43.67. Some apps always want you to put the sign on declinations. The aim here is to practice some of the "spherical geometry" equations from the book: Eq 1.1-5 gives the arc distance (B) between two stars If you have a plane triangle with sides A B C and opposite interior angles a b c, the law of cosine says C^2=A^2 + B^2 - 2 cos(c) A B Three stars in the sky form a spherical triangle and the equivalent statement is cos(C)=cos(A) cos(B) + sin(A) sin(B) cos(c) Given the sides of a spherical triangle we can use this to find the interior angles For this problem we employ the "Big Dipper" You can locate the Big Dipper on your SC002, and determine each star's RA & dec, but we seek greater accuracy. Given a real astronomical name, online apps like simbad will report "all" known astronomical data about the star (certainly RA & dec). But I've given you a common name...try "wiki big dipper" It should show a nice picture with labels giving common names e.g., Merak and corresponding greek letter -- beta -- (aka Bayer designation) Problem: go to simbad (if you're having trouble finding it, try: http://www.physics.csbsju.edu/370/photometry/Photo_jump.html Enter "Merek" as an identifier...I get about 40 different names for this star, record one that has at least six digits We could continue using simbad, but I think wiki will be easier. If you go down to the wiki section "Stars" there is a nice clickable table which allows you to get information about any Big Dipper star...record the RA & dec of Dubhe & Merak Use wiki or simbad to get and record the RA & dec of Polaris Problem: Merek, Dubhe, Polaris form a very skinny triangle (almost a straight line) in the sky. If you go the the "Guidepost" section of the wiki on the Big Dipper there is a diagram that claims the Polaris-Dubhe separation is about five times the Merek-Dubhe separation. Find those separations and report if the wiki is correct. Merek & Dubhe are known as "the Pointers;" the claim is if you extend the Merek-Dubhe line it will go through Polaris, that is Merek, Dubhe, Polaris make a very skinny triangle. How skinny is it? If it were actually a straight line the angle at Dubhe would be 180 deg. What is this angle actually? These trigonometric calculations get tiresome. On the class website there is a file spherical_geo.m which can be inserted into Mathematica to define functions that can do these calculations. The functions that end with "2" expect arguments in h.mmss format, the others require entering each sexagesimal digit separately. I hope you will examine the code and confirm it does what I said! --class2-- Find the offset today between sidereal time and our clock time (which is on daylight saving) using the books equations. (i.e., report *T-ZT for today) problems from Chapter 1: 1, 4, 5 next class we'll be moving on to Chapter 2, so read it! for those who have taken PHYS 339, this may seem elementary (no Lagrangian) web read: Tutorials for Astrophysics * Orbital Elements & Calculations http://www.physics.csbsju.edu/orbit/orbit.2d.html http://www.physics.csbsju.edu/orbit/orbit.3d.html * 3-body Problem: an example by movie http://www.physics.csbsju.edu/orbit/meissel.mp4 or webm (chrome) --class3-- problems from chapter 2: convenient units in the solar system are AU & years. Unit convert (G*Msun) to these units...you should get (2*pi)^2 i.e., meter->AU and seconds->years Plug that into the equation for omega and find: T (in years)=[a (in AU)]^(3/2) Test out this result for your favorite planet. If you use these units in the following problems you will be happier 2 (a dropped object moves in a straight line and hence has e=1, so what is a?) 3 6 (typo: v=1.2 AU/yr) --class4-- still in chapter 2, but moving to chapter 4 (skipping 3) on Wednesday problems from chapter 2: like #8, but with more details (but still not exactly realistic): In class we calculated the transfer orbit to Mars IF mars had a circular orbit. The real Mars orbit is eccentric, so there are two extreme cases: (A) transfer to Mars and join it at perihelion (B) transfer to Mars and join it at aphelion. The usual figure of interest is the change in velocity the rocket must achieve to change the orbit (delta V). To simplify, consider sending a satellite to Mars from a *circular* orbit with radius 1 AU coplaner with Mars' orbit (i.e., similar to Earth but circular and in the same plane as Mars). Determine the position/velocity of Mars at perihelion, and then figure the two delta V required to join Mars orbit at that perihelion from the circularized Earth orbit. (You will need to look up Mars orbit data on the web.) How long does it take to reach Mars? Determine the position/velocity of Mars at aphelion, and then figure the two delta V required to join Mars orbit at that aphelion from the circularized Earth orbit. How long does it take to reach Mars? wiki reports Halley's comet has eccentricity=e=.96714 semi-major axis=a=17.834 AU period=T=75.32 year What fraction of the time is the comet more than its semi major from the Sun? Hint: r=a requires cos(phi)=-e (Why?)...from phi calculate the eccentric anomaly u... from u calculate wt...the resulting time is the time since perihelion finally report the fraction of the period r>a Last year, for the first time, an asteroid that came from outside our Solar System was observed. see wiki: Oumuamua wiki reports (somewhat outdated) orbital data: Perihelion 0.25534 AU Semi-major axis −1.2798 AU (the negative sign just indicates hyperbolic; don't use it if you use my formulae) Eccentricity 1.19951 At the time JPL calculated the (hyperbolic) orbit and reported: Vinfinity=26 km/s calculate this yourself! "JPL solution indicates that one hundred years ago, the object was roughly 553 AU (83 billion km) from the Sun." The above sentence refers to time, which occurs in only one formula I've given you: e sinh(u)-u= omega t if you knew u you could get r: r=a(e cosh(u)-1) but there is no way to "solve" the equation to get algebra for u as a function of t I suggest FindRoot on mathematica or root finding on your calculator. Mercury_Tracking_Network_2.png shows the ground track of the 5th US manned space flight (Schirra, 3-Oct-1962). The flight was launched from FL heading towards Africa, went about 6 orbits and finished (as planned) with a splash-down in the Pacific. Note that the capsule did not fly over MN and that the orbits 'advanced' a bit (e.g., orbit 3 crossed the equator at about 10 deg and orbit 4 crossed at a bit less than 35 deg). Explain why these features were natural consequences of orbital mechanics. Why launch towards Africa? Why did it not go over MN? *Calculate* (based on orbital mechanics) the expected orbit advance and compare to the value visible in the ground track. FYI: the image: iss.gif shows a 3d rendering of the ISS orbit, but it may also help with the Mercury track. wiki reports the following orbit data: (Mercury-Atlas 8) Perigee 156 kilometers (84 nmi) Apogee 285 kilometers (154 nmi) Inclination 32.5 degrees Period 88.91 minutes What was the semi-major axis? What was the eccentricity? Assume you are 100 km behind the Space Station and you share the same circular orbit as the Space Station: 400 km above the Earth's surface. Recall orbit phasing. What delta v is required to catch up to the Space Station in one orbit. --class5-- chapter 4 swihart problems: 2,5,6 in the old folder, find 364t199.pdf; do #9 --class6--- important results from class: for x=h nu/kT mode: x=1.59362 median: x=2.35676 average: x=2.70118 total N = 8 pi (kT/hc)^3 2 Zeta[3] = 8 pi (kT/hc)^3 2.40411 = 20286794 T^3 (per m^3) total Energy/volume = 4 sigma/c T^4 Remark: if you were to look at the speed distribution of gas escaping from a hole, you'd not get Maxwell-Boltzmann as high speed atoms are more likely to escape than slow atoms. However this does not apply to light where all photons travel at c. However, instead of looking at the number of photons with frequency nu, folks commonly calculate the total energy coming out at frequency nu...this tacks on another factor of h nu to the distribution...thus escaping (at hole surface) energy flux=F=2 pi h nu^3/c^2/(exp(x)-1) chapter 4 swihart problems: 7,8 start chapter 5 next time (maybe) look up the number density of particles in our sea level atmosphere. At what temperature would the photons start to outnumber the particles? look up the sea level atmospheric pressure. Just as we derived for particles, bouncing photons produce a pressure: P = 4 sigma/(3c) * T^4 At what temperature would the photon pressure exceed the particle pressure? Remark: temperatures 100x this are common inside stars & 100^4=10^8 --class7--- tau_problem.pdf see 364t104.pdf #1 This problem asked students to find F_lambda, but our text stresses per nu values. SO Consider the number flux of photons "near" (you decide) the frequency that has lambda=.5E-6 m (green light) The quantity you seek reports the number of collected photons per second per m^2 of collecting area "at" a particular frequency (i.e., per bin size). Estimate the temperature of this star. The list of photon wavelengths (in micro meter) is 1000_lambda.txt. I think one of the first things you'll want to do is to copy & paste this data into a spreadsheet...talk to me if you're having problems doing that. --class8--- http://www.pdas.com/atmosTable2SI.html is a nice table of the U.S. Standard Atmosphere (1976) to 20 km Problem: WAPP (altitude,pressure) data for altitudes below 10 km; find the scale height (report units!) (There are too many columns in this table for WAPP to be happy; I used a spreadsheet to grab just the columns of interest) Make a semi-log plot of the results. The Lapse Rate is the rate at which temperature declines with altitude. WAPP (altitude,temperature) data for altitudes below 10 km; find the lapse rate (report units!) Make a linear plot of the results. In the model discussed in class we found: Lapse Rate = g/(constant pressure specific heat of air) Problem: Show that the units of the above work out to K/m Using the web find and cite the specific heat of dry air. (Note: specific heats are not exactly constants; they depend on temperature and pressure. Find a value that is appropriate for the atmosphere.) The wiki page on Lapse Rate has lots to say including the following: "While the dry adiabatic lapse rate is a constant 9.8 °C/km, the moist adiabatic lapse rate varies strongly with temperature. A typical value is around 5 °C/km," http://www.pdas.com/atmosTable1SI.html goes to 86 km WAPP all of these altitudes, report the scale height, and make a semi-log plot Our exponential pressure model incorrectly assumes a constant temperature, report your estimate of its usefulness Swihart chapter 5 #8 --class9--- after a bit more on chapter 5, we start chapter 6: stars Remark: an alternative source on magnitude/luminosity/temperature of stars is the Photometry Lab p.67-76, which is also online in the 370 web site. HD 145622 is a star (located in the middle of the back of the bowl of the Little Dipper) that you could see only under ideal conditions. At what rate (photons per second) are this star's photons entering your eye (assuming you are looking at the star)? Go to Simbad (Google "simbad astronomy" or use #8 in Photo_jump.html in the class web site) and retrieve this star's V magnitude. Using the basic filter data in Figure 4.3 p.75 in the Photometry Lab calculate the V flux; convert this energy-based flux to a photon count flux by assuming that the typical photon in the band carries the energy of the listed band-center wavelength photon. The resulting N_{\lambda} is the total count rate divided by the bin size (\Delta\lambda which is also tabulated) and detector area. Find the total number/sec of V-band photons entering an eye of diameter .5 cm. Note: the above is a measure of photons outside the atmosphere, not photons absorbed by the retina (which would be much less). Recent studies have show that humans can detect a "flash" that consists of a single absorbed photon. Swihart chapter 6: #5 & #6 The images taken in the Photometry Lab cover a region of 28'x28'. What fraction of the entire Celestial Sphere is covered by one image? (Lets call that 1/N) Thus if the region in the image is typical (unlikely) there will be N stars on the entire Celestial Sphere as bright or brighter than the brightest star in the image. According to what I found on the web there are 2,822 stars brighter than 5.5 mag. (This is about the dimmest star you could see unaided on a good night.) Using the results of #6, how many magnitudes dimmer than 5.5 must I go to find the typical bright star in the image. For comparison: about the best you can do aided by binoculars is mag 9. FYI: the actual increase in number with one magnitude is actually close to 3 (rather than 4). FYI2: the brightest star in the region I frequently study is mag 11 On average, how far apart (angle) are the mag 5.5 stars? It turns out that the general problem of equally spacing N particles on a sphere has no algebraic solution (this problem is named for JJ Thompson, the discover of the electron, as he was the first to discuss it in detail). However with large N each local patch must seem fairly flat, so the solution for a plane must approximately apply. The solution for a plane is called hexangular packing, and if D is the separation of the particles, the number of particles per area is 1/(D^2 sin(pi/3)) Using this result, find the typical angle between 2,822 equally spaced stars. --class10--- midterm exam in one week 364t104.pdf #6, #8 (Note: "Luminosity Class" V=main sequence, III=giants, I=supergiants) Swihart chapter 6: #1 & #2 --class11--- another version of 364t104.pdf #1 The file 1000_lambdaD.txt contains the wavelengths (in micro meter) of a 2 second exposure using our 30 cm diameter telescope. Note the large number of IR photons...we have a cooler star this time. Count the number of photons in the V band (see Photometry Lab p.75 for filter details), calculate F_lambda (W/(m^2*um)) in the V-band bin, and calculate the V mag. Do the same for the B filter. Calculate B-V and using Eq. 4.28 (Photometry Lab) find the star's temperature. Do note that F_lambda involves the total photon energy in the bin, not the photon count. According to the file the median wavelength (which relates to the median frequency) is 1.22095 micro meter. Using the results of class6, calculate the star's temperature using the median --class 12--- no homework help Sunday 2pm --class13-- mid-term exam one side of 8.5"x11" sheet formula sheet allowed I will provide the orbital mechanics equations that are file: merged.pdf chapters 1,2,4,5,6,7; photometry lab might also be worth reviewing --class14-- problems in chapter 8: 3 4 (note: mu changes between ionized H & He!) 5 (note: assume 10% of the Sun's mass is converted to He--remark "foe"=fifty-one ergs is the unit for supernovas) living flesh has an energy "production" (i.e., conversion: chemical PE->heat) of about 1 W/kg compare that to the average energy "production" of the Sun you might be interested to check out on-line stellar structure calculators: http://www.astro.wisc.edu/~townsend/static.php http://mesa-web.asu.edu/ --class 15-- to get the formula for "degeneracy pressure" wiki it! Find the mass density of Cu and the mass of a Cu atom. Use this to calculate the number of Cu atoms per m^3. Assume there are 2 valence electrons from each Cu that are "free". Calculate the electron number density, and the corresonding "degeneracy pressure". What is the result in atm? wiki "Fermi gas" and find (about half way down) the relation between the Fermi energy: E_F and the number of particles per volume, N/V If the Fermi energy is 0.1 mc^2, we must be getting close to the relativisitic case. For an electron gas find the N/V that is this limit. Comapre it to the Cu number density --class 16-- brief discussion of the Chandrasekhar/Eddington controversy http://www.pbs.org/wgbh/nova/blogs/physics/2012/01/the-chandrasekhar-limit-the-threshold-that-makes-life-possible/ isothermal.pdf https://www-nds.iaea.org/relnsd/vcharthtml/VChartHTML.html is a "Live Chart of Nuclides"...click on an isotope and get lots of stuff, including the Atomic Mass (in micro amu) Note that this is the mass of a neutral atom (i.e., it includes Z electrons). In a decay calculation you must account for those electrons also. The first reaction in the pp chain is; 1H+1H->2H+positron+neutrino Take the neutrino mass as zero; the positron mass exactly equals the electron mass. Find the energy released in MeV Carbon-14 decay is important for dating...find the energy released: 14C->14N+electron+neutrino remark: for this "beta- decay" of 14C, the "Live Chart of Nuclides" has a cell that reports the Q (energy release) for beta- decay of 14C. This should agree with your answer. --class17-- on to chapter 9 The following is really more of a magnitude problem then binary star problem so review how magnitudes work and how to get ratios of luminosities from magnitude differences Consider a binary eclipsing star system with stars "S" (the small-radius star) and "L" (the large-radius star). When L totally eclipses S, the combined light drops 0.2 magnitudes (i.e., the magnitude increases by 0.2). Find the luminosity ratio L_S/L_L. When S moves in front of L, the combined light drops 1 magnitude. What fraction of L is covered by S? The aim here is to see plots of radial velocity vs time for a binary star. Here are a couple of online applets that display similar plots: http://astro.unl.edu/classaction/animations/extrasolarplanets/radialvelocitysimulator.html http://physics.unm.edu/Courses/Rand/applets/spectroscopicBinaries.html The orbital mechanics (which we did in ch. 2), can be found again at: http://www.physics.csbsju.edu/orbit/orbit.2d.html A problem is that the orbit is best parameterized by the eccentric anomaly (u), and even that relationship is pretty complex. For any given u we can calculate the time (really w*t) from Kepler's equation: wt[e_,u_]=u-e Sin[u] where e is the eccentricity. The angle from perihelion (true anomaly) can also be found from u: phi[e_,u_]=2 ArcTan[ Cos[u/2], Sqrt[(1+e)/(1-e)] Sin[u/2] ] The velocity vector (in ASM units) is scaled by v0=2 Pi Sqrt[(m1+m2)/(a(1-e^2))] v[e_,u_]=v0 {-Sin[phi[u]],Cos[phi[u]]+e} v is actually the relative velocity: v1=(m2/(m1+m2))*v, v2=(m1/(m1+m2))*v, where a is the semimajor axis of the orbit and m1 and m2 are the masses of the stars. Note that these just change the scale of the orbit/speeds, the shape is determined from e and our view point. If the direction to Earth is at an angle i from the orbit's normal, and at an an angle w from the minor axis, the radial velocity as seen from Earth is: vr[e_,w_,i_,u_]=-v0 {-Sin[phi[u]],Cos[phi[u]]+e}.{Sin[w] Sin[i],Cos[w] Sin[i]} This is just the velocity vector dotted with a vector pointing toward Earth. We can now plot this radial velocity vs time using a ParametricPlot: ParametricPlot[{wt[e,u],vr[e,w,i,u]},{u,0, 4 Pi}] plotting two full cycles usually makes seeing the cycle clearer and of course we must put in values for e,w,i The class web site has binary.nb with the Mathematica commands to view these plots (a better alternative to using he applets.) The easiest way to see how a parameter affects a relationship is to animate the plot. Note that I have not bothered with a control to change the overall orientation of the orbit. Note that if the inclination is zero there is no radial velocity, and i=0 is the starting value. Give the orbit a ~45 deg inclination and a eccentricity of e=.5 See how the direction to Earth affects the radial velocity plot: See how the eccentricity affects the radial velocity plot: Your assignment: in the directory: http://www.physics.csbsju.edu/364/orbit/ find the files: orbit_X.gif and vr_X.gif where X is the last digit of your ID number. "Chi-by-eye" is the disparaging name for the process of fitting functions just by using trial and error comparison between the data and a plot using a trial set of parameters. Using an applet or Mathematica adjust the (e,i,w) parameters until you think you've got an approximate match (one set of e,i,w that works for both plots). Note that it's the shape you want to match not the values or orbit orientation. Note that the vr plot shape is not affected by i, just the range of vr (which is unknown). Report your best "fit" parameters (e,i,w) --class18-- on to chapter 10 Swihart ch.10 #1 --class 19 (=-4)--(following break) abel.pdf --class 20-- Friday 12 Oct LCDM.pdf Note: class 22 is the last class, Wednesday 17 Oct --class 21-- Monday 15 Oct hotBB.pdf don't forget class evaluation & final Final "due" Monday 22 Oct --class 22-- Wednesday 17 Oct Last Day of Class