1. Connection between math and astronomy? Mathematics is the language of astronomy! Motions on the sky Eclipses Comets Stellar motion Galaxies Expansion of the Universe

2. Nick Suntzeff B.S. in Math from Stanford ! Famous astronomer at Cerro-Tololo, Chile Physics Professor at TAMU starting next year Cerro-Tololo observatory Supernova in Centaurus A

3. Many times in the history of Astronomy careful attention to mathematical details led to great discoveries You don’t need to go very far, to use a sophisticated equipment, or to be a math genius Main points of this lecture: 1.Slight variations of the Earth’s orbit and Ice Ages 2.Tiny shift of the perihelion of Mercury and General relativity 3.Tiny wobbling of stars and discovery of extrasolar planets 4.Serendipitous discovery of pulsars 5.Faint supernovae and accelerating universe …

4. Newton’s law of gravitation Gravity: the main force in the Universe

5. Orbits due to the force ~ 1/r 2 are elliptical Sun Earth (Eccentricity greatly exaggerated!) Perihelion = position closest to the sun Aphelion = position furthest away from the sun Orbit of the Earth is almost circular Perihelion: 147 million km; Aphelion: 152 million km

6. Seasons 1.Seasons are NOT caused by varying distances from the Earth to the Sun 2.The primary cause of seasons is the 23.5 degree tilt of the Earth's rotation axis with respect to the plane of the ecliptic. Note: the Earth is actually closest to the Sun in January 4! The Seasons in the Northern Hemisphere Perihelion: 147.09 × 10 6 km; Aphelion: 152.10 × 10 6 km 23.5 o tilt

7. We experience Summer in the Northern Hemisphere when the Earth is on that part of its orbit where the N. Hemisphere is oriented more toward the Sun and therefore: 1.the Sun rises higher in the sky and is above the horizon longer, 2.The rays of the Sun strike the ground more directly. Likewise, in the N. Hemisphere Winter the hemisphere is oriented away from the Sun, the Sun only rises low in the sky, is above the horizon for a shorter period, and the rays of the Sun strike the ground more obliquely.

8. Myr ago Puzzle: Ice Ages! Occur with a period of ~ 250 million yr Cycles of glaciation within the ice age occur with a period of 40,000 yr Most recent ice age began ~ 3 million yr ago and is still going on!

9. Last Glacial Maximum: 18,000 yr ago 32% of land covered with ice Sea level 120 m lower than now

10. Ice Age: Cause Theory: climate changes due to tiny variations in the Earth’s orbital parameters –Precession of the rotation axis (26,000 yr cycle) –Eccentricity (varies from 0.00 to 0.06 with 100,000 and 400,000 yr cycles) –Axis tilt (varies from 24.5 o to 21.5 o with 41,000 yr cycle Milutin Milankovitch 1920

12. 26,000 yr cycle

14. Varies from 0.00 to 0.06 (currently 0.017) Periodicity 100,000 and 400,000 yr Eccentricity cycle modulates the amplitude of the precession cycle

15. As a result, the flux of solar radiation received by the Earth oscillates with different periodicities and amplitudes This triggers changes in climate Our Earth makes a complicated motion through space, like a crazy spaceship

16. f1 f2 f3 f1+f2 f1+f2+f3 Adding oscillations with different phases and incommensurate frequencies f1 = sin[2 t + 1] f2 = 0.7 sin[3.1 t + 2.4] f3 = 1.3 sin[4.5 t + 0.3]

17. Adding Milankovitch cycles of solar irradiation for 65 degree North latitude (Berger 1991) Note the last peak 9,000 years ago when the last large ice sheet melted

18. Very good agreement!

19. Are these effects enough to explain the Ice Ages??? Other factors? Volcanic winters, impacts, … 71,000 yr ago: eruption of Mount Toba (Sumatra) 2,800 km 3 of material thrown in the atmosphere Instant ice age? Meteorite impacts; Mass extinctions

20. Are we alone in the Universe?? Are there any Earth-like planets in habitable zones around stars?

21. Star-Birth Clouds in M16 (Eagle Nebula). J. Hester and P. Scowon (Arizona St. Univ.), November 2, 1995. Taken with NASA Hubble Space Telescope, WFPC2 We know there should be planets around other stars. How to find them???

22. M > 2000 M 

23. JUPITER SHINES WEAKER THAN THE SUN: 1 000 000 000 TIMES (visible light) 50 000 TIMES (infrared) 5 000 TIMES (mm and sub-mm)

24. JUPITER OBSERVED FROM THE NEAREST STAR 0.1” DISTANT FROM THE SUN „DROWNED” IN SUNSHINE !!

25. POPULAR VIEW

26. REALITY X = CENTER OF MASS SYSTEM VIEWED POLE-ON (RARE)

27. Center of Mass (SLIDESHOW MODE ONLY)

28. ...BUT THE PLANET CANNOT BE SEEN MOTIONS OF THE STAR BETRAY ITS PRESENCE !

29. X EARTH X JUPITER 150 000 000 km 30 km/s 450 km 9 cm/s 780 000 000 km 13 km/s 750 000 km 13 m/s

30. 2010 2000 2005 1995 1990 2015 2020 0.002” MOTIONS OF THE SUN VIEWED FROM A STAR 30 LIGHT YEARS AWAY 0.002’’ IS THE ANGULAR SIZE OF A MAN ON THE MOON OR A STANDARD NEWSPAPER FONT 300 KM AWAY Unobservable!

31. STELLAR WOBBLE RECEDING: REDDER APPROACHING: BLUER

32. The Doppler Effect The light of a moving source is blue/red shifted by  / 0 = v r /c 0 = actual wavelength emitted by the source  Wavelength change due to Doppler effect v r = radial velocity Blue Shift (to higher frequencies) Red Shift (to lower frequencies) vrvr

33. 1 Angstrom = 10 -8 cm

34. PLANET DETECTION DUE TO STELLAR WOBBLE -50 -100 0 100 50 m/s 0321 days K P i to the observer normal to the orbit VV K = V  sin i

35. Only the function of masses and inclination angle can be measured

36. ANOTHER EFFECT: TRANSIT PLANET IN FRONT OF THE STAR

37. TIME BRIGHTNESS TRANSIT 1% LIGHT CURVE

38. First direct detection of radiation from extrasolar planet! Spitzer Infrared Telescope

39. Over 100 planets discovered

40. EXPECTED: NEARLY CIRCULAR ORBITS BIG PLANETS FAR AWAY FROM THE STAR NO PLANETS BIGGER THAN JUPITER DISCOVERED: STRONGLY ELONGATED ORBITS BIG PLANETS VERY CLOSE TO THE STAR MANY PLANETS BIGGER THAN JUPITER

41. Planetary system of  And Solar system 0.06 AU 4.5 days 0.75 M J 2.5 AU 3.5 years 4 M J 0.85 AU 242 days 2 M J 0.39 AU 89 days 0.73 AU 228 days 1 AU 1 year 1.54 AU 1.9 years Source: Harvard-Smithsonian CfA

42. Habitable zones

43. Signs of life in the spectrum:

45. The advance of the perihelion of Mercury

46. Orbits due to the force ~ 1/r 2 are elliptical Sun Mercury Perihelion = position closest to the sun Aphelion = position furthest away from the sun Perihelion: 46 million km; Aphelion: 70 million km

48. Mercury's perihelion precession: 5600 arcseconds/century Newtonian perturbations from other planets: 5557 arcseconds/century GR correction: 43 arcseconds/century