Hubble s Law PHYS 1301

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1 1 PHYS 1301 Hubble s Law Why: The lab will verify Hubble s law fr the expansin f the universe which is ne f the imprtant cnsequences f general relativity. What: Frm measurements f the angular size and wavelengths f light bserved we calculate the distance and speed f recessin f several galaxies, pltting the results n a speed vs. distance graph t determine the Hubble cnstant and hence the age f the universe. Hw: We will use existing telescpe data n the emissin and absrptin spectra f several galaxies tgether with images t determine their angular size. Backgrund In the 1920's, Edwin P. Hubble discvered a relatinship that is nw knwn as Hubble's Law. It states that the recessinal velcity f a galaxy (hw fast it is mving away frm us) is prprtinal t its distance frm us: ν = H d (1) where v is the galaxy's velcity (in km/sec), d is the distance t the galaxy (in megaparsecs Mpc; 1 Mpc = 1 millin parsecs, 1 parsec = 3.26 light years), and H is the prprtinality cnstant called "The Hubble Cnstant". Hubble's Law implies that a galaxy mving away frm us twice as fast as anther galaxy is twice as far away. In rder t precisely determine the value f H, we must determine the velcities and distances t many galaxies, preferably thse extremely far away. If the universe has been expanding at a cnstant speed since its beginning, cmparing equatin (1) t speed = distance/time, we see that the universe's age wuld simply be t = 1/H. This will allw yu t estimate the age f the universe. Velcity v : This can be calculated using the Dppler effect. This effect means that electrmagnetic waves received frm a surce mving away frm the bserver are shifted t lwer frequency f cmpared t thse cming frm the same surce if it were at rest relative t the bserver. Because the speed f light c = 3 x 10 5 km/s is cnstant, this

2 2 means that the wavelength f the wave (its clr) defined by λ = c/f is shifted t a lnger (redder) wavelength. The rati f this shift t the true rest wavelength is called the redshift z = (λmeasured - λtrue)/ λtrue (2) λtrue is the wavelength f light as emitted frm a certain surce (usually ht gas) in a lab n Earth, while λmeasured is the wavelength received at a telescpe frm the same kind f surce in a fast-mving galaxy. The thery f the Dppler Effect then shws that the recessinal velcity f a galaxy is ν= c z (3) Galaxies emit light f many different wavelengths, but certain wavelengths, called emissin and absrptin lines, tend t be very prminent and can be used t easily determine the redshift due t mtin f the surce. Example: An absrptin line is measured in the lab at 5000 Å (1 Ångstrm (Å) = m). When analyzing the electrmagnetic spectrum f a certain galaxy, the same line is fund at 5050 Å. Knwing the speed f light, we calculate that this galaxy is receding at v = (50/5000) x c r apprximately 3000 km/s. Distance d : We will assume that all galaxies f the same type are the same physical size s, n matter where in the Universe they are. By perspective, bjects further away lk smaller they have a smaller angular size a when viewed thrugh a telecspe and this can be used t calculate the distance d t the galaxy frm the gemetrical frmula valid fr small angles a d = s / a (4) The typical size s has been determined (using ther methds) with nearby spiral galaxies such as Andrmeda, Triangulum, Messier 81, etc. and is n average s = 22 kilparsecs (22 kpc r 22,000 parsecs r abut 72,000 light years) We will assume this is the typical size f all spiral galaxies, measure their angular size a frm telescpe images, then calculate their distance d frm us using equatin (4). Prcedure

3 3 G t the Galaxy List at First yu will measure angular size and calculate the distance t several galaxies. 1. Chse a galaxy frm the list. The images are negatives, s that bright bjects, such as stars and galaxies, appear dark. There may be mre than ne galaxy in the image; the galaxy f interest is always the ne clsest t the center. T measure the angular size a, click n ppsite ends f the galaxy, at either end f the lngest diameter. Be sure t measure all the way t the faint uter edges, therwise yu will dramatically underestimate the size f the galaxy, and intrduce a systematic errr. The cmputer will autmatically calculate the angular diameter in milli-radians (yu can ignre pixel crdinates x1,y1, x2, y2). 2. Repeat step 1 fr anther fur galaxies and recrd yur results fr a in a data table 3. Calculate the distance t each f the galaxies in Mpc (Megaparsecs) using equatin (4) with milli-radians units fr a d [Mpc] = s [kpc] / a [milli-rad]. Nw yu will measure the redshift and calculate the velcity f yur chsen galaxies. Click n a galaxy's spectrum link and yu will see a graph f the full visible light spectrum f the galaxy similar t that shwn t the right. This shws hw the relative intensity (brightness) f the light seen frm the galaxy depends n the wavelength (clr) f the light.

4 4 Enlarged prtins f the same spectrum are als shwn, in the vicinity f sme prminent spectral features. Fr each galaxy yu will use three well-knwn surces f light, called CaK, CaH, and Hα, t get three independent estimates f the galaxy s velcity. CaK and CaH are absrptin spectral lines, meaning their intensity is exceptinally lw, while Hα is an emissin spectral line, meaning the intensity is exceptinally high. 4. The rest wavelengths f CaK, CaH, and Hα will be given in the header captin f the graphs. Recrd the λtrue values f these wavelengths. The small vertical bars near the lwer left crners f the graphs mark the values f these rest wavelengths. These wavelengths becme red-shifted t larger values frm a galaxy mving away frm us. They can be lcated by searching the graphs fr nearby prminent dips in the intensity in the case f CaK and CaH, while fr Hα the feature will be a nearby prminent peak in the intensity. Ask the instructr if yu are nt sure what t lk fr. 5. Fr each galaxy in yur table, measure the shifted wavelength λmeasured fr each f the 3 spectral lines by clicking at the middle f the crrespnding feature (dip r peak) in the intensity graphs. 6. Fr each galaxy, use equatin (2) t calculate the z values frm each f the different spectral lines. If they seem cmpatible, fr each galaxy average yur z values. If yur z values fr a galaxy are nt cmpatible re-check yur calculatin r re-measure the wavelengths (yu may have picked the wrng feature). 7. Using equatin (3) and yur average z values, calculate the recessinal velcity f each galaxy in km/s. Graph yur results fr v and d t find the Hubble cnstant by fllwing these steps: 1. Plt yur data fr each galaxy with distance (in Mpc) n the x-axis, and velcity (in km/s) n the y-axis. 2. Draw the steepest reasnable line and the shallwest reasnable line n the graph that seems t fit the pints n the graph. These lines must bth pass thrugh the rigin (0,0) (why?) A gd fit will have pints scattered either side f the line. 3. Measure the slpes f these lines (rise/run); these slpes are yur upper and lwer bunds n the value f the Hubble cnstant H.

5 5 Age f the Universe Yur H has been measured in funny units km/s/mpc that astrnmers use, s t find the age f the universe in years, we must cancel the distance units (1 Mpc = 3.09 x km) and cnvert secnds t years (1 yr = 3.16 x 10 7 s) s that Age [years] = 3.09 x / (3.16 x 10 7 H) Calculate an upper and lwer bund n the age f the universe frm yur values f H. Cnclusins Write a paragraph summarizing the key things yu learnt frm yur data, quting data t supprt yur cnclusins.

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