A Mathematicians Adventures

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A Mathematicians Adventures in the Universe
Connection between math and astronomy?
Mathematics is the language of astronomy!

• Motions on the sky
• Eclipses
• Comets
• Stellar motion
• Galaxies
• Expansion of the Universe

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B.S. in Math from Stanford !
Famous astronomer at Cerro-Tololo, Chile
Physics Professor at TAMU starting next year
Nick Suntzeff
Cerro-Tololo observatory
Supernova in Centaurus A
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Main points of this lecture
Many times in the history of Astronomy careful
attention to mathematical details led to great
discoveries
You dont need to go very far, to use a
sophisticated equipment, or to be a math genius

• Slight variations of the Earths orbit and Ice
  Ages
• Tiny shift of the perihelion of Mercury and
  General relativity
• Tiny wobbling of stars and discovery of
  extrasolar planets
• Serendipitous discovery of pulsars
• Faint supernovae and accelerating universe

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Gravity the main force in the Universe
Newtons law of gravitation
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Orbits due to the force 1/r2 are elliptical
Earth
Perihelion position closest to the sun
Sun
Aphelion position furthest away from the sun
(Eccentricity greatly exaggerated!)
Orbit of the Earth is almost circular
Perihelion 147 million km Aphelion 152 million
km
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Seasons

• Seasons are NOT caused by varying distances from
  the Earth to the Sun
• 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.

23.5o tilt
The Seasons in the Northern Hemisphere
Note the Earth is actually closest to the Sun in
January 4!
Perihelion 147.09 106 km Aphelion 152.10
106 km
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• 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
• the Sun rises higher in the sky and is above the
  horizon longer,
• 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.

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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!

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Last Glacial Maximum 18,000 yr ago 32 of land
covered with ice Sea level 120 m lower than now
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Ice Age Cause

• Theory climate changes due to tiny variations in
  the Earths 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 24.5o to 21.5o with 41,000
  yr cycle

Milutin Milankovitch 1920
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26,000 yr cycle
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• 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

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Our Earth makes a complicated motion through
space , like a crazy spaceship
As a result, the flux of solar radiation received
by the Earth oscillates with different
periodicities and amplitudes This triggers
changes in climate
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Adding oscillations with different phases and
incommensurate frequencies
f1 sin2 t 1 f2 0.7 sin3.1 t 2.4
f3 1.3 sin4.5 t 0.3
f1
f2
f3
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Adding Milankovitch cycles of solar irradiation
for 65 degree North latitude
Note the last peak 9,000 years ago when the last
large ice sheet melted
(Berger 1991)
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Very good agreement!
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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 km3 of material thrown in the
atmosphere Instant ice age?
Meteorite impacts Mass extinctions
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Are we alone in the Universe??
Are there any Earth-like planets in habitable
zones around stars?
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We know there should be planets around other
stars. How to find them???
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
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JUPITER SHINES WEAKER THAN THE SUN
1 000 000 000 TIMES (visible light)
50 000 TIMES (infrared)
5 000 TIMES (mm and
sub-mm)
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JUPITER OBSERVED FROM THE NEAREST STAR
0.1 DISTANT FROM THE SUN
DROWNED IN SUNSHINE !!
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POPULAR VIEW
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REALITY X CENTER OF MASS SYSTEM
VIEWED POLE-ON (RARE)
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Center of Mass
(SLIDESHOW MODE ONLY)
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THE PLANET CANNOT BE SEEN
...BUT
MOTIONS OF THE STAR BETRAY ITS PRESENCE !
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EARTH
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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!
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STELLAR WOBBLE RECEDING REDDER
APPROACHING BLUER
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The Doppler Effect
The light of a moving source is blue/red shifted
by
Dl/l0 vr/c
l0 actual wavelength emitted by the source Dl
Wavelength change due to Doppler effect vr
radial velocity
Blue Shift (to higher frequencies)
Red Shift (to lower frequencies)
vr
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1 Angstrom 10-8 cm
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PLANET DETECTION DUE TO STELLAR WOBBLE
K V?sin i
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Only the function of masses and inclination
angle can be measured
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ANOTHER EFFECT TRANSIT PLANET IN FRONT OF
THE STAR
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First direct detection of radiation from
extrasolar planet!
Spitzer Infrared Telescope
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Over 100 planets discovered
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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
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Planetary system of u And
0.85 AU 242 days 2 MJ
2.5 AU 3.5 years 4 MJ
0.06 AU 4.5 days 0.75 MJ
0.73 AU 228 days
1 AU 1 year
0.39 AU 89 days
1.54 AU 1.9 years
Solar system
Source Harvard-Smithsonian CfA
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Habitable zones
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Signs of life in the spectrum
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The advance of the perihelion of Mercury
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Orbits due to the force 1/r2 are elliptical
Perihelion position closest to the sun
Mercury
Sun
Perihelion 46 million km Aphelion 70 million km
Aphelion position furthest away from the sun
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Mercury's perihelion precession 5600
arcseconds/century Newtonian perturbations from
other planets 5557 arcseconds/century GR
correction 43 arcseconds/century
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