Astronomy Exam on 10/2/19 - Study Guide & Homework Set #6

1. PHYS 3380 - Astronomy The first exam will be next Wednesday (10/2/19) at the regular class time. It is closed book but you may bring in one 8 1/2 X 11 inch “cheat sheet” with writing on both sides. I expect the exam to be done in pencil. You will need a calculator. It will cover everything I have covered in class up to and including Wednesday, 9/18, i.e., up to but not including Light.

2. PHYS 3380 - Astronomy • Homework Set #6 • 9/30/19 • Due 10/9/19 • Chapter 6 • Review questions 3 and 6 • Problems 4 and 13 • Extra Problems • At a temperature of 5800 K, hydrogen atoms have typical random speeds of about 12 km/s. Assuming that spectral line broadening is simply the result of atoms moving away or towards us at this random speed, estimate the thermal width (in nanometers) of the 656.3-nm H line. • Vulcan has an escape velocity 1.4 times that of Earth. It radius and density are, respectively, 1.2 and 1.5 times that of earth. It’s surface temperature is 298 K. What is it’s ESI? • Venus can be as bright as apparent magnitude -4.7 when at a distance of about 1 AU. How many times fainter would Venus from a distance of 1 pc? What would its apparent magnitude be? Assume Venus has the same illumination phase from your new vantage point. • Light travels from air into an optical fiber with an index of refraction of 1.44.  (a)  In which direction does the light bend?  (b)  If the angle of incidence on the end of the fiber is 22o, what is the angle of refraction inside the fiber?  (c)  Sketch the path of light as it changes media.

3. PHYS 3380 - Astronomy Spectral Line Shapes • In classical picture of the atom as the definitive view of the formation of spectral lines: • - spectral lines should be delta functions of frequency • - appear as infinitely sharp black lines on stellar spectra • However, many processes tend to broaden these lines - lines develop a characteristic shape or profile • - quantum mechanical effects - natural or radiation broadening • according to Heisenberg's uncertainly principle, product of the uncertainty in the measurement of energy, ΔE, and time Δt is: • - results in a natural spread of photon energies around the spectral line. The longer an excited state exists (Δt), the narrower the line width so that metastable states can have very narrow lines. • - intrinsic to atom itself • - natural width ~0.001 - 0.00001 nm

4. PHYS 3380 - Astronomy Spectral Line Broadening • - Zeeman effect • the splitting of a spectral line into several components in the presence of a static magnetic field. • attributed to the interaction between the magnetic field and the magnetic dipole moment associated with the orbital angular momentum. • used by astronomers to measure the magnetic field of the Sun and other stars • - collisions with neighboring particles • potential of charged particles interacts with that of the atomic nucleus which binds the orbiting electrons. • perturbs the energy levels of the atom in a time-dependent fashion - broadens the spectral line. • - motions of the atoms giving rise to the line • - macroscopic - highly ordered, i.e. stellar rotation • microscopic - random, i.e., thermal motions, turbulence • - Doppler broadening

5. PHYS 3380 - Astronomy Collisional Broadening Occurs when atoms absorb or emit photons while colliding with other atoms, ions, or electrons - potential of charged particles interacts with that of the atomic nucleus which binds the orbiting electrons. - perturbs the energy levels of the atom - can absorb slightly wider range of wavelengths - dependent on temperature and density of gas Balmer 434.0 nm line (H) from two A1 stars (same temperature) - differences in width due to differences in density of gas

6. PHYS 3380 - Astronomy Doppler Broadening Caused by motions of individual atoms in gas - some moving towards observer (blueshift), some moving away (redshift) - some moving faster, some moving slower - broadens spectral line - dependent on temperature Cool gas Hot gas

7. PHYS 3380 - Astronomy Measuring Rotational Velocity Doppler shift can be used to tell us how fast an object is rotating: As an object rotates, light from side rotating toward us is blueshifted - light from side rotating away from us is redshifted. Spectral lines appear wider - the faster it rotates, the wider are the spectral lines.

8. PHYS 3380 - Astronomy Extrasolar Planets • Planets which orbit other stars are called extrasolar planets. • Over the past century, we have assumed that extrasolar planets exist, as evidenced from our science fiction. • We finally obtained direct evidence of the existence of an extrasolar planet in the year 1995. • A planet was discovered in orbit around the star 51 Pegasi. • To date: • 4055 Confirmed Planets around 3014 Stars • 667 Systems with Multiple Planets • 4717 KeplerCandidates and Confirmed planets FYI: Most of the information I give here came from the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu) and the Kepler spacecraft website (http://kepler.nasa.gov/). There is a lot more information on those sites (and others) if you are interested

9. PHYS 3380 - Astronomy Detecting Extrasolar Planets • Can we actually make images of extrasolar planets? • No, this is very difficult to do. • The distances to the nearest stars are much greater than the distances from a star to its planets. • The angle between a star and its planets, as seen from Earth, is too small to resolve with our biggest telescopes. • A star like the Sun would be a billion times brighter than the light reflected off its planets. • As a matter of contrast, the planet would be lost in the glare of the star. • Improved techniques of interferometry may solve this problem someday.

10. Detection Methods Astrometry- precisely measure star's position in the sky and observe the ways in which that position changes over time - gravitational influence of the planet causes the star to move in a tiny orbit about common center of mass Radial velocity or Doppler method - variations in the speed with which the star moves towards or away from Earth deduced from the displacement in the parent star's spectral lines due to the Doppler effect. Pulsar timing - slight anomalies in the timing of observed radio pulses used to track changes in the pulsar's motion caused by the presence of planets. Transit method - observed brightness of the star drops by a small amount as a planet crosses in front of its parent star's disk. Gravitational microlensing- gravitational field of a star acts like a lens, magnifying the light of a distant background star. Possible planets orbiting the foreground star cause detectable anomalies in the lensing event light curve.

11. Detection Methods Circumstellar disks - disks of space dust are detected because dust absorbs ordinary starlight and re-emits it as infrared radiation. Features in dust disks may suggest the presence of planets. Eclipsing binary – planet detected by finding variability in light curve minima as it goes back and forth - most reliable method for detecting planets in binary star systems. Orbital phase – observing planetary orbital phases - depends on inclination of the orbit. By studying orbital phases scientists can possibly calculate particle sizes in the atmospheres of planets (using albedo calculations). Two planets have been discovered by Kepler using this method. Polarimetry - stellar light becomes polarized when it interacts with atmospheric molecules, which could be detected with a polarimeter. So far, one planet has been studied by this method.

12. PHYS 3380 - Astronomy Atacama Large Millimeter/submillimeter Array (ALMA) images of circumstellar disks.

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14. PHYS 3380 - Astronomy Radial Velocity Doppler shift allows detection of slight motion of star caused by orbiting planet

15. PHYS 3380 - Astronomy Determining Star’s Velocity Animation

16. PHYS 3380 - Astronomy • A plot of the radial velocity shifts forms a wave. • Its wavelength tells you the period and size of the planet’s orbit. • Its amplitude tells you the mass of the planet. Doppler shift in spectrum of star 51 Pegasi - shows presence of large planet with orbital period of about 4 days.

17. PHYS 3380 - Astronomy Determining Planet Mass and Orbit Animation

18. PHYS 3380 - Astronomy Remember - Doppler shift only tells us radial motion. If plane of orbit perpendicular to our line of sight - no shift seen. If we view it from edge on, maximum Doppler shift seen. Orbit generally tilted at some angle - star’s full speed not measured. So mass derived from Doppler technique is minimum possible. If changing velocity and varying position in sky measured (as in one case - Gliese 876) orbital tilt can be determined and mass measured accurately. Gliese 876 is only about 15 LY away.

19. PHYS 3380 - Astronomy Planetary Transit • The Doppler technique yields only planet masses and orbits. • Planet must eclipse or transit the star in order to measure its radius. • Size of the planet is estimated from the amount of starlight it blocks. • We must view along the plane of the planet’s orbit for a transit to occur. • transits are relatively rare • They allow us to calculate the density of the planet. • Initially, all the extrasolar planets detected had Jovian-like densities. Since then, because of the Kepler spacecraft, we have detected a number of rocky planets. • Method used by Kepler spacecraft Planetary Transit Animation

20. PHYS 3380 - Astronomy Planets Discovered by Kepler Spacecraft • The Kepler Mission was “specifically designed to survey a portion of our region of the Milky Way galaxy to discover dozens of Earth-size planets in or near the habitable zone and determine how many of the billions of stars in our galaxy have such planets.” • uses transit method • Launched March 2009 • 2345 confirmedplanets • 400 Earth sized (< 1.25 RE) • 847 - 1.25 RE < R < 2 RE • 361 within habitable zone • On November 4, 2013, astronomers reported, based on Kepler space mission data, that there could be as many as 40 billion Earth-size planets orbiting in the habitable zones of sun-like stars and red dwarf stars within the Milky Way Galaxy. 11 billion of these estimated planets may be orbiting sun-like stars. The nearest such planet may be only 4 light-years away.

21. Confirmed Planets:2345Planet Candidates: 4765 Kepler Mission The first page of the Kepler catalog

22. PHYS 3380 - Astronomy Confirmed planet statistics

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25. PHYS 3380 - Astronomy Potentially Habitable Exoplanets Habitable Size     Habitable Orbit M (ME) R (RE)   d (AU) Teq (K) S (So) P (days) Min 0.1 0.5   inner 0.75 294 1.78 237 max 10 2.5   outer 1.84 187 0.29 910 Subterran (Mars-size) TerranSuperterran (Super-Earth) Total 1 20 34 55 subterran = 0.1 — 0.5 ME or 0.4 — 0.8 RE, terran = 0.5 — 5 ME or 0.8 — 1.5 RE,superterran = 5 — 10 ME or 1.5 — 2.5 RE

26. PHYS 3380 - Astronomy If you look at the habitable planet archives, you will note that the equilibrium temperature of the Earth is listed as about 255K (288K with greenhouse effect) instead of the 279K in my calculations last class period. This is because I chose to ignore albedo in the calculation to simplify them. Albedo is the reflctivity of an object. 0 is all light absorbed (black body) and 1 is all light reflected. The Earth’s albedo is about 0.3. So the calculations using albedo are:

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28. List of exoplanets more likely to have a rocky composition and maintain surface liquid water (i.e. 0.5 < Planet Radius ≤ 1.5 Earth radii or 0.1 < Planet Minimum Mass ≤ 5 Earth masses, and the planet is orbiting within the conservative habitable zone). PHYS 3380 - Astronomy

29. Earth Similarity Index (ESI) where xi is a planetary property (e.g. surface temperature), xio is the corresponding terrestrial reference value (e.g. 255 K), wi is a weight exponent, n is the number of planetary properties, and ESI is the similarity measure. The weighting exponents are used to adjust the sensitivity of the scale and equalize its meaning between different properties. Earth-like planets can be defined as any planetary body with a similar terrestrial composition and a temperate atmosphere. As a general rule, any planetary body with an ESI value over 0.8 can be considered an Earth-like planet. This means that the planet is rocky in composition (silicates) and could have an atmosphere suitable for most terrestrial vegetation including complex life.  Planetary Property Reference Value  Weight Exponent   Mean Radius   1.0 Eu   0.57   Bulk Density   1.0 Eu   1.07   Escape velocity   1.0 Eu   0.70  Surface Temperature   288 K   5.58

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32. PHYS 3380 - Astronomy • A potentially habitable world orbiting Proxima Centauri, closest star to Earth, was discovered in 2016 using using the radial velocity method • ProximaCentauri is a cool red-dwarf slightly older than the Sun  • Proximab • minimum mass 1.3 times that of Earth • suggests a rocky composition • radius between 0.8 and 1.4 Earth radii • 11.2-day orbit • receives 70% the energy Earth receives from the Sun • equilibrium temperature 227 K • ESI of 0.87 • Probably tidally locked • illuminated side might be too hot - dark side too cold for liquid water or life • thick atmosphere or a large ocean, though, could regulate the temperatures across the planet • Most red dwarfs are very active • strong magnetic fields, flares, and high UV and X-ray fluxes • may lead to the atmospheric and water loss, high radiation • magnetic field could potentially provide a shield • Either a lucky find or these worlds are more common than previously thought • Close enough to Earth for detailed studies in the next years • Perhaps a goal for projects like StarShot

33. PHYS 3380 - Astronomy The K2 Mission In 2013, Kepler lost the second of its four reaction wheels, rendering it incapable of maintaining precision pointing in three dimensions. Engineers devised a remarkable solution: using the pressure of sunlight to stabilize the spacecraft so it could continue to do science. The K2 mission began in Mar 2016

34. PHYS 3380 - Astronomy • The Transiting Exoplanet Survey Satellite (TESS) • Launched on April 18, 2018, aboard a SpaceX Falcon 9 rocket. • Searching for exoplanets using the transit method like Kepler. • Will survey 200,000 of the brightest stars near the sun to search for transiting exoplanets. • Will survey the entire sky over the course of two years by breaking it up into 26 different sectors, each 24 degrees by 96 degrees across. • The stars TESS will study are 30 to 100 times brighter than those the Kepler mission and K2 follow-up surveyed, which will enable far easier follow-up observations with both ground-based and space-based telescopes. • will also cover a sky area 400 times larger than that monitored by Kepler. • So far has found 1184 candidates and 29 confirmed exoplanets

35. PHYS 3380 - Astronomy Lenses and Mirrors

36. PHYS 3380 - Astronomy Properties of Light Law of Reflection - Angle of Incidence = Angle of reflection Law of Refraction - Light beam is bent towards the normal when passing into a medium of higher Index of Refraction. Light beam is bent away from the normal when passing into a medium of lower Index of Refraction. index of Refraction - Inverse square law - Light intensity diminishes with square of distance from source. ISNS 4371 Phenomena of Nature

37. PHYS 3380 - Astronomy Law of Reflection  Normal  Angle of incidence () = angle of reflection () The normal is the ray path perpendicular to the mirror’s surface.

38. PHYS 3380 - Astronomy Geometry of a Concave Mirror Focus Principal axis Vertex Focal length Center of curvature - the center of the circle of which the mirror represents a small arc Principal axis - a radius drawn to the mirror surface from the center of curvature of the mirror - normal to mirror surface Focus - the point where light rays parallel to principal axis converge; the focus is always found on the inner part of the "circle" of which the mirror is a small arc; the focus of a mirror is one-half the radius Vertex - the point where the mirror crosses the principal axis Focal length - the distance from the focus to the vertex of the mirror

39. PHYS 3380 - Astronomy Index of Refraction As light passes from one medium (e.g., air) to another (e.g., glass, water, plexiglass, etc…), the speed of light changes. This causes to light to be “bent” or refracted. The amount of refraction is called the index of refraction.

40. PHYS 3380 - Astronomy Refraction Imagine that the axles of a car represent wave fronts. If the car crosses from a smooth to a rough surface at an angle, one tire of the axle will slow down first while the other continues at normal speed. With one tire traveling faster the other, the car will turn in the direction of the slow tire. This is how refraction works.

41. PHYS 3380 - Astronomy AIR Car GLASS / WATER Slower Propagating Speed ( Sand /Gravel)

42. PHYS 3380 - Astronomy AIR Car GLASS / WATER Slower Propagating Speed ( Sand / Gravel )

43. PHYS 3380 - Astronomy AIR NORMAL GLASS / WATER Slower Propagating Speed

44. PHYS 3380 - Astronomy NORMAL LIGHT BENDING TOWARDS THE NORMAL AIR LIGHT RAY GLASS / WATER Slower Propagating Speed

45. PHYS 3380 - Astronomy n2 NORMAL LIGHT BENDING TOWARDS THE NORMAL AIR n1 Snell's Law ( Next Slide ) GLASS / WATER Slower Propagating Speed

46. PHYS 3380 - Astronomy Snell's Law Where: VL1 is the longitudinal wave velocity in material 1. VL2 is the longitudinal wave velocity in material 2. Snell's Law describes the relationship between the angles and the velocities of the waves. Snell's law equates the ratio of material velocities VL1 and VL2 to the ratio of the sine's of incident and refracting angles.

47. PHYS 3380 - Astronomy Snell's Law n=(c/v) where : C is the velocity of light and v is the velocity of light in that medium where 1 and 2 are the angles from the normal of the incident and refracted waves, respectively. n1, n2 are indices of refraction of the two media respectively.

48. PHYS 3380 - Astronomy GLASS / WATER ( Sand / Gravel ) Slower Propagating Speed Car AIR

49. PHYS 3380 - Astronomy GLASS / WATER ( Sand / Gravel ) Slower Propagating Speed Car AIR

50. PHYS 3380 - Astronomy GLASS / WATER ( Sand / Gravel ) Slower Propagating Speed Car AIR