Download Mathematics and Cosmos

Connection between math and
astronomy?
Mathematics is the language of
astronomy!
• Motions on the sky
• Eclipses
• Comets
• Stellar motion
• Galaxies
• Expansion of the Universe
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
Main points of this lecture:
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
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
…
Gravity: the main force in the Universe
Newton’s law of gravitation
Orbits due to the force ~ 1/r2 are
elliptical
Perihelion = position
closest to the sun
Earth
Sun
(Eccentricity greatly exaggerated!)
Orbit of the Earth is almost circular
Perihelion: 147 million km; Aphelion: 152 million km
Aphelion
= position
furthest
away
from the
sun
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.
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
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.
Puzzle: Ice Ages!
Myr ago
• 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!
Last Glacial Maximum: 18,000 yr ago
32% of land covered with ice
Sea level 120 m lower than now
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 24.5o to
21.5o with 41,000 yr cycle
Milutin Milankovitch 1920
26,000 yr cycle
• 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
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
f1 =
sin[2 t + 1]
f2 = 0.7 sin[3.1 t + 2.4]
f3 = 1.3 sin[4.5 t + 0.3]
Adding oscillations with different
phases and incommensurate
frequencies
1
f1
f1+f2
0.5
2
2
4
6
8
10
1
-0.5
-1
2
1
f2
4
6
8
10
8
10
-1
0.5
-2
2
4
6
8
10
f1+f2+f3
-0.5
f3
-1
2
1
1
0.5
2
-1
2
-0.5
-1
4
6
8
10
-2
4
6
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)
Very good agreement!
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
Are we alone in the Universe??
Are there any Earth-like planets in habitable zones around stars?
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
M > 2000 M
JUPITER
SHINES WEAKER THAN THE SUN:
1 000 000 000 TIMES
50 000 TIMES
5 000 TIMES
(visible light)
(infrared)
(mm and sub-mm)
JUPITER
OBSERVED FROM THE NEAREST STAR
0.1” DISTANT FROM THE SUN
„DROWNED” IN SUNSHINE !!
POPULAR
VIEW
REALITY
X = CENTER
OF MASS
SYSTEM VIEWED
POLE-ON (RARE)
Center of Mass
(SLIDESHOW MODE ONLY)
THE PLANET CANNOT BE SEEN
...BUT
MOTIONS
OF THE STAR
BETRAY
ITS PRESENCE !
450 km
9 cm/s
150 000 000 km
30 km/s
X
EARTH
750 000 km
13 m/s
JUPITER
X
780 000 000 km
13 km/s
MOTIONS
OF THE SUN
VIEWED FROM
A STAR 30
LIGHT YEARS
AWAY
2020
1995
2010
1990
0.002’’ IS THE
ANGULAR
SIZE OF A
MAN ON THE
MOON OR A
STANDARD
NEWSPAPER
FONT 300 KM
AWAY
2015
2000
0.002”
Unobservable!
2005
STELLAR
WOBBLE
RECEDING:
REDDER
APPROACHING:
BLUER
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
Blue Shift (to higher
frequencies)
vr
Red Shift (to lower
frequencies)
Dl = Wavelength
change due to
Doppler effect
vr = radial
velocity
1 Angstrom = 10-8 cm
PLANET DETECTION DUE TO STELLAR WOBBLE
100
m/s
P
normal to
the orbit
i
to the observer
50
K
0
V
-50
-100
0
1
2 days 3
K = Vsin i
Only the function of masses and inclination angle
can be measured
ANOTHER
EFFECT:
TRANSIT
PLANET
IN FRONT
OF THE STAR
TRANSIT
BRIGHTNESS
LIGHT
CURVE
1%
TIME
First direct detection of radiation from
extrasolar planet!
Spitzer Infrared Telescope
Over 100 planets discovered
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
Planetary system of u And
0.85 AU
242 days
2 MJ
0.06 AU
4.5 days
0.75 MJ
0.39 AU
89 days
0.73 AU
228 days
1 AU
1 year
2.5 AU
3.5 years
4 MJ
1.54 AU
1.9 years
Solar system
Source: Harvard-Smithsonian CfA
Habitable zones
Signs of life in the spectrum:
The advance of the perihelion of Mercury
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
Mercury's perihelion precession: 5600 arcseconds/century
Newtonian perturbations from other planets: 5557 arcseconds/century
GR correction: 43 arcseconds/century