Middle and High School investigations in astronomy

1. Measuring what you can't touch - from the size of the Earth to the structure of the solar system, black holes and the expansion of the Universe—session 3 Middle and High School investigations in astronomy

2. Plan for This Session • Background and Exploration of Activity 2, subactivity 2 (HS)– Cepheid variables and the distance to galaxies • Background and exploration of Activity , 3, subactivity 1 (Both MS and HS)—escape velocities and gravity • Break • Background and exploration of Activity 3, subactivity 2 (both MS and HS) Kepler’s laws and orbits. • Lunch? • Background and exploration of Activitiy 3, subactivity 3 (HS onlyl)—Black holes • Background and exploration of Activity 3, subactivity 4 (HS only) – Hubble’s law and the expansion of the Universe • Break • Discusssion and evaluations.

3. Activity 2, subactivity 1– The inverse square law. You are all familiar with the concept. Light sources that are farther away appear fainter.

4. How light spreads Light from a source that emits in all directions (isotropically) spreads out uniformly. At any given point in time, it covers a sphere whose radius increases with time (it’s increasing at the speed of light). That same light therefore is spread out over a larger area. The surface area of a sphere is Area=4pr2 Therefore, the energy per area falls b y the same factor Each unit of area has less light passing through it when it is further from the source than when it is closer to the source.

5. Standard candles Of course, the relation between flux (which you can measure) and distance doesn’t work if you don’t know the luminosity. That means you need to have Standard Candles In the first sub-activity, you will be working with a standard candle (standard lightbulb?)– In the second sub-activity, you will study a class of stars that can be used as standard candles.

6. Activity 2, sub-activity 2—putting it into practice. Now that we’ve learned about the inverse square law, we can put it to use to measure the distances to stars. First, we need to find some standard candles. In the first activity, we learned that stars aren’t all alike. So we need some special stars. We need Cepheid variables

7. Cepheid variables—what are they? Cepheid variables are stars that pulsate. They are constantly growing and shrinking, with very regular periods. As the stars pulsate, they get alternately more and less luminous The flux at Earth keeps changing

8. Why do Cepheids pulsate:more information than you need The stars pulsate because of opacity changes. In the interior of the star, there is a layer where the helium gas in the star is doubly ionized. The ionized gas absorbs more light, and this heats the gas. The hot gas expands, making the star bigger. But as gas expands, it cools. When the gas cools, it also becomes less ionized. This decreases the light absorption, and the gas cools so fast it contracts, contracting the star. The contracting gas heats, and re-ionizes, and the cycle starts all over again….

9. Cepheid variables—why do they matter? Cepheids aren’t all the same luminosity—some are brighter than others. But, they do have an interesting property, discovered by Henrietta Leavitt in 1908 during her study of the Magellanic clouds: There is a relation between the period of the Cepheids and their luminosity Photo Credit: AAVSO

10. The period-luminosity relation (1) What Leavitt found was that in the SMC, stars with longer period were brighter. The data is on the right. The x-axis plots the logarithm of the pulsation period, in days (so 0.0 is 1 day, and 2.0 is 100 days). The y-axis plots the apparent magnitude of the stars in the photographs. Magnitudes is a system of measuring the flux from astronomical objects astronomers have carried around with them since Hipparchus in the 1st century BC. Hipparchus called the brightest stars in the sky 1st magnitude stars, the next brightest 2nd magnitude, and so on until the faintest were 6th magnitude. In modern magnitudes, the Sun is -26.7, and the faintest galaxies ever detected are at +30 or so The actual data from Leavitt’s 1912 paper The “modern” magnitude system assigns a magnitude of 0.0 to some flux (traditionally the flux of Vega), and then each magnitude difference corresponds to a factor of 2.512 in flux (5 magnitudes is a factor of 100)

11. The period-luminosity relation (2) The apparent magnitude (m) measures the flux– which depends on distance! There is an equivalent magnitude, called the absolute magnitude (M), which measures the luminosity—actually it measures the flux you would measure if the object is at a fixed distance. The difference between m and M depends on the distance—if d is in parsecs: m – M = 5 log(d) – 5 The Sun has M~+5 in this system, while the most luminous quasars discovered have M~-24. The actual data from Leavitt’s 1912 paper The period-luminosity relation has been calibrated using the parallax measurements of the Hipparcos satellite: M = -2.81 log(P) – 1.43 The Luminosity can be measured from the period!

12. Hubble and the distances to galaxies. The Cepheid variables were the key to measuring the distances to galaxies: • They are bright enough to be seen in nearby galaxies • Their luminosity can be determined from the period—they are standard candles. In 1924, Edwin Hubble used the then biggest telescope in the world (the Mt. Wilson 100” telescope) to measure cepheids in a nearby galaxy-- Andromeda

13. HST and the distances to galaxies. Credit: ESO • The Hubble space telescope was built partly to measure Cepheid variables in nearby galaxies (it was one of its “key” projects). Putting a telescope in space makes finding individual stars much easier (less blurring). With the space telescope, we have been able to measure Cepheids in tens of galaxies. You will be using the data collected by HST on one of the galaxies, M100. (The 100th object in Charles Messier’s album of fuzzy things that were not comets! M100 is a beautiful spiral galaxy. From the ground, individual stars are hard to spot. With Hubble Space telescope, it’s just possible. Credit: HST

14. The cepheid light curves. You will measure periods and magnitudes for 12 Cepheids measured by Freedman et al. (1994) The period will be used to infer M, then the distance will be calculated from m – M = 5 log(D) - 5

15. Possible extensions. • Once you have the distances to galaxies, this can be the starting point for investigating the properties of galaxies (types, rotation, stellar populations) • Another possible place to go is to investigate how we measure distances to even farther objects (where single cepheids can’t be seen)—this leads to the topic of the distance ladder.

16. Activity 3– gravity enters the picture • The first two activities on both the MS and the HS side deal primarily with measuring distances and sizes. • To understand many of the relations between astronomical objects in the solar system and beyond, you need to understand the force that dominates on astronomical scales—gravity.

17. Gravity from the Earth out As usual, our strategy is to work from the Earth out. First we’ll study the escape velocity on Earth and on other planets to discover how it depends on mass and radius, and how it differs from the velocity needed to enter a circular orbit.

18. The escape velocity If you throw a ball in the air, it will go up, stop and then fall back. If you throw it faster, it goes higher up, the ball goes further—but it doesn’t leave the earth. You might have even learned a formula for the height (y) a ball reaches given an initial upward velocity Vy: y = yinitial + Vy2/(2g), where g is the acceleration due to gravity, 9.8 m/s2 Is this formula always true? If so, then there will always be a maximum height an object will reach! But, that can’t be the case— otherwise how would the rovers have gotten to Mars?

19. Gravity and distance. The answer to this puzzle is that the force of gravity is not constant—it varies as you go up from the surface of Earth. Once you get far enough from the center of the Earth, the gravitational force gets weaker. At some critical velocity, the force of gravity falls fast enough that the object manages to just barely escape the gravity of Earth. This is the critical velocity.

20. Energy and the escape velocity—some physics. The escape velocity is the initial velocity Vi at which an object will reach zero final velocity when the distance from Earth is infinite. This means that the kinetic energy difference between beginning and end is DKE = m/2 (Vf2-Vi2)= -mVi2/2 The energy is always conserved, so this must be equal to the negative change in gravitational potential energy: -DPE = GMearthm(1/Rf – 1/Ri) = -GMearthm/Ri energy conservation says DE = DKE + DPE = 0, or Vi2 = 2GMearth/Ri When the object is launched from the surface of the Earth, Ri = Rearth.

21. Experience with escape velocities We don’t have real rockets to test whether the relation is valid. Instead, we will rely on simulations. Simulations have an advantage—you can change not just the rocket initial velocity, but the Mass and the radius of the planet. In this subactivity, you will be investigating how the escape velocity depends on M and R, to verify the equation we just derived.

22. Activity 3, subactivity 1: Escape Velocities. Explore. • As you work through the activities: • What do you like? • What don’t you like? • What are some gaps? • Errors/difficulties? • Work through all pages up through the engage sections. • Can you determine the relation between the escape velocity and the velocity in a circular orbit?

23. Activity 3, subactivity 2: Orbits and Kepler’s laws. • In the last sub-activity, you found that the speed required to put the rocket in a circular orbit is 1/sqrt(2) of the escape velocity. This is also easy to understand with a little bit of physics. We need two concepts: • Motion in a circle is accelerated, with acceleration acirc = -v2/R • The acceleration due to Gravity is agrav=-GM/R2

24. Putting the two together: V2 = GM/R (just like the escape velocity except for a factor of two). Aside– this is a general law—the circular orbit always has KE = -PE/2 (this is called the Virial theorem). There are some other laws of orbits that help in determining the structure of the solar system– Kepler’s laws.

25. Kepler’s first law The first of Kepler’s laws is also the hardest to derive from the law of gravity (it’s only done in the honors intro physics class at Brown). The law is: Law 1: All (closed) orbits are ellipses, with the center of mass of the system at one of the foci

26. How to make an ellipse To make an ellipse, you need a piece of string and two fixed points (the foci of the ellipse). The ellipse is described by the eccentricity and the major, or semi-major axis.

27. More magic– Kepler’s second law Kepler’s second law says how objects move in the elliptical orbits. It’s at first magical 2nd Law: objects move in their orbits so as to sweep equal areas in equal times. What does this mean?

28. Consequences of Kepler’s second law. For this to work, the object has to move faster when it’s close to the Sun and slower when it’s farther. (This turns out to be just a consequence of the conservation of angular momentum). The second law allows you to describe the motions of planets throughout their orbit

29. The scaling of the Solar System—Kepler’s third law If the second law describes the motion of planets in their orbit, the third goes further—it relates all the orbits to each other. 3rd Law: In an orbit, the square of the orbital period is proportional to the cube of the semi-axis. How planets move depends on their distance from the Sun.

30. Deriving Kepler’s third law from the circular orbit. The third law (at least for circular orbits) is just a consequence of what we already derived. V2 = GM/R for a circular orbit; but the velocity is just V = distance traveled/time= 2pR/P (P is the period) So 4p2R2/P2 = GM/R Rearranging: P2 = 4p2/(GM) R3 Very important note—the constant depends on the Massof the thing orbited—for the Sun it’s the mass of the Sun, but for orbits around other objects it’s the mass of the the object at the center…

31. We can make it simpler… The constant is messy, can we pick units in which it is simpler? Pick P to be in years, and R in astronomical units. For the Earth, P=1,R=1. What is the constant in these units? 1! In these units: P2 = R3/M (M is in units of the mass of the Sun!) Rather than deriving the relations—have the students experience them via the simulation….

32. Activity 3, subactivity 3—No escape: black holes (HS only) Now that we’ve gotten a feel for Kepler’s laws and the escape velocity, let’s apply them to something much more exotic than the solar system– black holes! We’ll see, that only a small extension of what we already did will allow us to learn of the event horizon, and to determine the mass of the giant black hole in the center of the Milky Way.

33. The Speed of light The one concept we do need to understand black holes is that there’s a cosmic speed limit. The speed of light (in a vacuum) is as fast as anything can move (this is one of the key underpinnings of Einstein’s theory of special relativity, which is experimentally verified every day in particle colliders). Light travels extremely fast (300,000 km/s), But space is big.

34. Aside– telescopes as time machines. Because space is so big, when we look at distant objects, we see into the past. Not our own past, but no matter. This is extremely important for undersanding cosmology. It allows us to look into the past of the Universe and study how it evolves! Without this time machine, we would not have as much confidence in the big bang model. But with it, we can show that the Universe in the past was really smaller, hotter, and denser!

35. The limit of escape velocities—the event horizon O.k., now we can combine the escape velocity and the speed of light. Let’s leave the mass fixed, and make things smaller--- then the escape velocity grows, until at some radius: c2 = 2GM/R– or R = 2GM/c2 This defines the event horizon. c is a big number, so R is small.

36. Why a black hole is black Because nothing can have a velocity greater than c, once something drops inside the event horizon, it can’t ever escape. Not even light can escape—it’s a one way trip. Matter falling in is forced to move ever inwards, towards the central singularity—where the density is so great that we do not yet have a physical theory that we know works. The radius of the event horizon is called the Schwarzschild radius.

37. Black holes: maligned? Black holes are usually portrayed as dangerous and sinister, sucking matter into them. The truth is not like that. Away from the event horizon, the effect of a black hole is just like that of any other object of the same mass “Gravity doesn’t care what the mass is”. You can have stable orbits around black holes just like anything else…

38. Orbits around a black hole Kepler’s laws are just as valid for orbits around black holes (at least for radii significantly bigger than the event horizon). P2 = R3/M (P is the period in years, R is the semi-major axis in AU, and M is the mass in solar masses)

39. Practical example—what’s hiding in the galactic center? The central region of our galaxy is hidden from our view by all the dust and gas in the galaxy (we are looking down the plane of the disk). But in the infrared or X-rays, the central region can be revealed. In the center is a very bright X-ray source, and lots and lots of stars. Chandra image of the center of the Milky Way For years, astronomers have been studying the central region in the infrared.

40. In the galactic center—the data The galactic center is a dynamic place—the stars move! As they move, they appear to orbit around a central, dark point.

41. The orbits of the stars. The stars have orbits that have periods measured in years. In particular, the closest stars have completed an orbit since the monitoring started.

42. The physical size of the orbit from the angular size From the picture below, we can measure the angular size of the semi-major axis. The methods we studied in the first activity can give us the physical size: R = D * (A/206,265) Where D is the distance, and A is the angle (in arcseconds). The distance to the center of the Milky Way has been measured to be D= 8,000 parsecs. What is R? (actually, since 1 parsec = 206,265 AU, R (in AU) = D * A

43. Kepler’s third law and the mass of the Galaxy’s central black hole. From the figure below, we can also estimate the period. Now, Kepler’s third law states that P2 = R3/M So, M = R3/P2 in solar masses A 4 million solar mass central object--- A supermassive black hole lives in the Milky Way center!

44. Possible extensions—more on black holes • Once you have the escape velocity, you can talk about what happens as you fall into black hole (animations) • Cover how black holes in binary systems are found • Spinning black holes and energy mining in sci-fi stories. (how are black holes treated in sci-fi?) • Are black holes really black?(Hawking radiation) For more ideas, you can look at websites like http://www.pbs.org/wgbh/nova/blackhole/

45. Activity 3, sub-activity 4: Hubble’s law and the expansion of the Universe. For the last sub-activity, we come back to the distances, but also with a touch of gravity underlying it all. Let’s tackle the expansion of the Universe. There are two topics covered in the sub-activity: Hubble’s law, and how to visualize the expanding Universe.

46. Distances of galaxies—Cepheids and beyond To recover Hubble’s law we need two pieces– the distances to galaxies and the redshift. We’ve talked about measuring the distance to one galaxy using Cepheid variables. Distances to other galaxies can be measured using other standard candles, like Type Ia supernovae---distances are hard and time-consuming. They will be given.

47. The doppler effect and velocities The other measurement we need is the velocity of the galaxies. This is measured by the Doppler effect. The Doppler effect is the shift in the measured frequency (or wavelength) of a wave due to relative motion of emitter and observer. Dl/l = vsource/vwave

48. Aside– spectroscopy(see also PS investigation 4!) To measure the Doppler effect, you need to know the emitted wavelength or frequency. How do we know? Atoms of different types (when not in a solid or liquid) emit light only at very specific wavelengths. This spectral signature can be used to determine the composition, but it also gives the emitted wavelength.

49. The doppler effect for light—the redshift Starting about 100 years ago, the spectra of galaxies have been recorded. These spectra show a mix of continuum (light at all wavelengths) and light at specific wavelengths. The surprise – almost all the galaxies had their lines shifted to longer wavelengths—redshifted!

50. Hubble’s law! In 1929, Edwin Hubble decided to plot the distances to galaxies he had been measuring using the Cepheids with the redshifts of these galaxies. The rest is history. V= Ho D (velocity is proportional to distance)