Computational Physics Project: Determining Hubble's Constant - Prof. Donald G. Luttermoser, Study Guides, Projects, Research of Physics

Download Computational Physics Project: Determining Hubble's Constant - Prof. Donald G. Luttermoser and more Study Guides, Projects, Research Physics in PDF only on Docsity! PHYS-4007/5007: Computational Physics Course Computer Project #1 Instruction Packet Dr. Donald G. Luttermoser Department of Physics and Astronomy East Tennessee State University Fall 2008 . In the 1920’s, Edmund Hubble compared magnitudes of galaxies to their redshifts and found that the fainter the galaxy, the bigger the redshift =⇒ Hubble’s Law. The more distant a galaxy (i.e., fainter galaxies), the larger the redshift hence recession velocity =⇒ The Universe is Expanding. Hubble’s Law mathematically is simply a linear relation between the distance, d to a galaxy and its radial (line-of-sight) velocity, vr: vr = H◦ d . The slope of the line representing this linear relation has become known as Hubble’s constant and is represented by the symbol H◦ (the “◦” means current value). Up until just the last decade, H◦ was not accurately known (various studies put in the range 50–100 km/sec/Mpc) =⇒ its actual value is of the utmost importance! The primary reason the Hubble Space Telescope was built was to determine an accurate value for H◦. It will be your job to determine H◦ from actual galactic spectra! Another parameter that is often used in cosmology is the redshift, z, which is defined as z ≡ ∆λ λ◦ = vr c , where we also include the nonrelativistic (vr  c) form of the Doppler Effect (∆λ/λ◦ = vr/c). We also can write a more general expression for the redshift, z = ∆λ λ◦ = √ 1 + vr/c√ 1 − vr/c − 1 . where, here, we have included the relativistic (i.e., vr < ∼ c) correction for the Doppler Effect. Rewriting this relativistic formula, we can express velocity as a function of redshift: vr c = (z + 1)2 − 1 (z + 1)2 + 1 , You will note that there is no way for a galaxy’s velocity to exceed that of light when using the relativistic form of the Doppler Effect. Prior to Hubble’s discovery, the Universe was assumed to be static, infinite, and eternal. From this assumption, it was reasoned that we should light in every direction we look =⇒ Olber’s Paradox. But Hubble showed that the Universe is expanding (hence not static) and is not eternal, hence the assumptions in Olber’s paradox are invalid. Since the galaxies are moving apart from each other, there must have been some time in the past when the 3 galaxies were fairly close together. This thought gave rise to the Big Bang Theory — it had a beginning! As a result of this, light gets redshifted out of the visible band. Also, as we look out, we look back in time. We cannot look infinitely far out since, sooner or later, we will see the Big Bang (and indeed we do see the Big Bang as the cosmic microwave background radiation )! Let’s ask the question, how long ago did this happen, that is, when were all of the galaxies close together? Initially, we will treat this question in a very simplified way. Using simple mechanics, an object will travel a distance d going at a certain constant velocity v in a time period T given by d = v T . Here, we are using d as the distance to a galaxy, v as the velocity that the galaxy is traveling, and T is the time since the galaxy started from the origin. But from Hubble’s Law, v = H◦ d, so T = d v = d H◦ d = 1 H◦ From this simple exercise, we need to point out that the Universe is actually younger than that due to gravity slowing down the expansion over time — this time corresponds to a max- imum age of the Universe. This Hubble Time can be expressed in terms of the measured Hubble constant via T◦[years] = 978 × 109 H◦[km/s/Mpc] . If T = 15 billion years, then a galaxy farther than 15 billion ligh years away, we will never seen at the present time since light would not have had enough time to reach us =⇒ 15 billion light years would be the size of the observable universe. Continuing on with our Newtonian description of the Universe, for Hubble’s Law (the ob- served kinematic world model ) to be true, galaxies in the Universe have to be distributed homogeneously (matter is uniformly distributed in space) and isotropically (the Universe looks the same in every direction) on a large scale. E.A. Milne and W.H McCrea (1934) extended this, at first purely kinematic model, so as to make it a Newtonian cosmol- ogy. They investigated the motions of a medium (the gas of galaxies, i.e., treat galaxies as point particles) that can take place in accordance with Newtonian mechanics if one demands throughout homogeneity, isotropy, and irrotational motion. Consider at time t, a galaxy at distance R(t), then according to Newton’s law of gravitation, this galaxy is attracted by the mass within a sphere of radius R given by mass M = (4π/3)R3 ρ(t), where ρ(t) is the mass-density at the instant considered. Thus, the equation of motion of this galaxy is determined by setting the force of motion equal to the gravitational force: d2R dt2 + GM R2 = 0, 4 where the mass M is constant. Multiplying each term in this equation by Ṙ = dR/dt, it is then possible to easily integrate this equation and obtain the energy equation : 1 2 ( dR dt )2 − GM R = h, where h is the integration constant, or Ṙ2 R2 − 8π 3 Gρ(t) + kc2 R2 = 0, in which we have written −h = kc2/2 in anticipation of a comparison with relativistic calculations. From this equation, it can be seen that the Universe cannot remain static if ρ > 0 at any time. We can define the current Hubble constant as H◦ = Ṙ◦/R◦, where we denote present time t = t◦ by a subscript ◦. In reality, Hubble’s constant is not constant at all, and should be expressed as the Hubble parameter, H(t), where H◦ is the current value of H(t), or H(t) = Ṙ(t) R(t) . Using this in the equation of motion we wrote above, we can write a new equation of motion based on the Hubble parameter: dH dt + H2 = −4π 3 Gρ(t) . Likewise, from the energy equation, we can write H(t)2 = 8π 3 Gρ(t) − kc 2 R2 . As can be seen, the value of Hubble’s “constant” changes over time and this change is a function of the density of the Universe at that given time. For a complete characterization of a model universe a the current epoch, we need, besides H◦, a second variable that describes the deceleration of the Universe due to its mass M . This is the so-called deceleration parameter: q◦ = − ( R̈◦ R◦ ) / ( Ṙ◦ R◦ )2 = − R̈◦ R◦ H2◦ = 4πGρ◦ 3H2◦ . This formula relates the acceleration R̈◦ to a uniform acceleration which would lead to the observed velocity R◦H◦ at distance R◦ in the Hubble time T◦ = H −1 ◦ , starting from zero 5 ngc3077 4700 4800 4900 5000 5100 5200 Wavelength (A) 0.00 0.10 0.20 0.30 0.40 F lu x ( 1 0 -1 3 e rg / s/ cm 2 / A H -b et a [O I II ] [O I II ] ngc3077 6400 6500 6600 6700 Wavelength (A) 0.0 0.5 1.0 1.5 F lu x ( 1 0 -1 3 e rg / s/ cm 2 / A H -a lp h a [N I I] Figure 1: Emission lines in galaxy NGC 3077. 8 ngc224 3900 3950 4000 4050 4100 Wavelength (A) 0.0 0.5 1.0 1.5 2.0 F lu x ( 1 0 -1 3 e rg / s/ cm 2 / A C a II K C a II H ngc224 5700 5800 5900 6000 Wavelength (A) 0 1 2 3 4 F lu x ( 1 0 -1 3 e rg / s/ cm 2 / A N a I D 2 N a I D 1 Figure 2: Absorption lines in galaxy NGC 244. 9 The total wavelength range covered by the spectra is typically 1230-7500 Å. The UV spectra are from the Atlas of archival IUE spectra by Kinney et al. (1993, ApJS, 86, 5). The optical spectra were observed with the Kitt-Peak 0.9-m telescope with the infrared spectrograph during September 1991 and July 1992. The spectral resolution is 5-6 Å for the IUE spectra and 10 Å for the Kitt Peak spectra. Your galaxy spectrum reading procedure (rdgalspec) should be called from your driver pro- cedure (hubble) and return the galaxy name (which you will need to retrieve from the file name), the wavelength vector, and the flux vector. Note that some of the data in these files have zero flux values for certain wavelengths (and some have zero wavelengths too). Make sure you trim off any flux and wavelength values of zero before returning the flux and wavelength vectors (i.e., arrays) to hubble. Many times you will have scientific data that is expressed in very large and/or very small numbers. When plotting such numbers, for appearance, it is often best to multiply the data by a scale factor and show that scale factor in the label title for a given plot (as shown in both Figures 1 and 2 where the flux on the y-axis has been multiplied by 1013, hence the flux is labeled as 10−13 erg/s/cm2/Å). In the driver procedure hubble, I set the scale factor with the routine galflscale which I have supplied to you on the Project #1 web page. 4.4 Plotting the Spectra. You will need both to make plots onto the terminal screen and to a hardcopy, such as an encapsulated postscript file (see mkproj1figs.pro for details as to how to do this. I suggest that you use the sampleplot.pro (rename it of course) as your template for doing this and refer to mkproj1figs.pro for helpful hints in order to accomplish your goal. Since you will have both absorption line and emission line spectra, I recommend that your program show you the whole spectrum first, then you interactively tell the code which type of spectrum you currently have. I have included a nice little GUI widget procedure that I wrote, spectype.pro, that will help you select the spectrum type. Once you determine the spectrum type, you will load the appropriate lines (emission or absorption), that is, their IDs and their wavelengths into the spectrum analysis portion of the code. 10 7 Appendix: FAQ The following is supplemental information that you might find useful for this project. I display this information as answers to posed questions. 1. How many plots should I include in my final manuscript? At least three plots should be included. Give me one example of fitting a Gaussian profile to one emission spectral line of one galaxy spectrum that you downloaded and a second plot of a Gaussian fit to an absorption line from one of the galaxy absorption spectrum. This can be done by using a solid line for the spectrum line and a dashed-line with asterisk markers (use LINE=2, PSYM=-2 keywords in your OPLOT command) for the Gaussian fit. The third plot should be your final results with velocity (in km/s) on the vertical axis and distance (in Mpc) on the horizontal axis. On this graph, plot your final linear function as a solid straight line with the PLOT command and the individual galaxies with asterisks using the PSYM=2 keyword in the OPLOT command. Plotting the v and d uncertainties for each galaxy point would be a nice touch, but don’t waste a lot of time trying to do this. 2. How do I calculate the total uncertainty of the Hubble constant? Essentially you are just calculating Eq. (V-48) in your notes with the partials determined in the same manner that they were determined in Eqs. (V-38) through (V-40). There are three sets of uncertainties that have to be combined to calculate the lone uncertainty in H. a) Uncertainties in wavelength (i.e., the exact center of the Gaussian fit). Let’s say, for example, that each spectrum has 3 lines that we are measuring. The GAUSSFIT function in IDL will give us 3 different line centers for these lines in the A[1] parameter (e.g., A1[1], A2[1], A3[1], that is, the median of the Gaussian is the predicted line center). Now let’s say that these lines have rest wavelengths of W1, W2, and W3. The Doppler effect gives (A[1] – W) / W = V / C, where V is the radial velocity and C is the speed of light (make sure you use a fairly accurate value for C = 2.997925 × 105 km/s). Now as you can see, you will get 3 different velocities from your 3 different DELTAWAVE = A[1] – W (= delta lambda) with Vi = (DELTAWAVE / Wi) ∗ C, “i” indicates each line in a given spectrum. Your velocity for that galaxy will be Vave = (V1 + V2 + V3) / 3, and your uncertainty in velocity for that galaxy is VSIGMAj = MAX( | Vi – Vave | ). Now each galaxy will have its own uncertainty for velocity. At this point, calculate the total “normalized” uncertainty for velocity SVNORM = SQRT( SUM [(VSIGMAj / Vave-j)∧2] ), where “j” represents each galaxy and SUM simply means add all of these terms together (one could make use of the TOTAL function in IDL for this) and ∧ is the IDL math symbol for raise to this power. 13 b) Uncertainty in distance (given in the galdist.txt file). Once again, we want a total nor- malized uncertainty here, so SDNORM = SQRT( SUM [(DISTj / UNC DISTj)∧2] ). Note that we are using the square root of the sum of the squares because the data is uncorrelated. c) Uncertainty in the slope of the fitted line. The method of least squares is handled with the IDL function called LINFIT. An example call to LINFIT would be MFIT = LINFIT(DIST, Vave, SIGMA=SLEASTSQ). LINFIT will return MFIT which is a two-element array containing the y-intercept (MFIT[0]) and the slope (i.e., Hubble’s constant, MFIT[1]) of the fitted straight line. In this command, DIST is an input array containing all of our DISTj data for all “j” galaxies and Vave is an input array containing all of the average velocity measurements (which has to be a one-to-one correlation with the data in DIST). The SLEASTSQ variable passed through the keyword SIGMA will be a two-element array upon output containing the uncertainties in the y-intercept (SLEASTSQ[0]) and the slope (i.e., Hubble’s constant uncertainty from the fit, SLEASTSQ[1]). After this function, one would need to normalize this uncertainty with SHNORM = SLEASTSQ[1] / MFIT[1]. d) The Total Uncertainty in H◦. The total normalized uncertainty is then SNORM = SQRT( SVNORM∧2 + SDNORM∧2 + SHNORM∧2 ) and the final uncertainty in your calcu- lated Hubble’s constant is given by DELTAH = SNORM ∗ MFIT[1]. 3. Why are the individual uncertainties in Hubble’s constant normalized in the above answer? Since each parameter in the uncertainty calculation has different units, one must use the normalization technique described above. Then multiply this normalized uncertainty by the actual calculated Hubble constant to convert this normalized uncertainty to the proper units. 14