Transcript Cosmic Surveyor

How the simple measurements of distances and positions of
planets and stars have led to great discoveries in science
2005 is the World Year of PHYSICS
100th anniversary of Albert Einstein’s
“miraculous year” of 1905
March 1905: the quantum nature of
light
May 1905: Brownian motion shows
the existence of atoms and
molecules
June 1905: Special Relativity as a
theory of space, time, and motion
General Relativity
Developed in 1907-1915 in close collaboration with
mathematicians: Grossmann, Hilbert, Levi-Civita
... in all my life I have not laboured nearly so
hard, and I have become imbued with great
respect for mathematics, the subtler part of
which I had in my simple-mindedness regarded
as pure luxury until now.
Marcel Grossmann
David Hilbert
Tullio Levi-Civita
Measuring distances: a surveyor’s method
A surveyor’s method in space:
measuring the parallax effect
an apparent shift of the object on the sky when viewed from
different locations
One needs larger baseline than the distance between the eyes to
measure the distance to cosmic objects!
In 1672 Giovanni Cassini together with Jean Richter (1630-1696) made
parallel observations of the Mars parallax in Paris (France) and Cayenne
(French Guiana, N. coast of South America)
They were also able to determine the solar parallax as ~ 9 arcseconds
and find the distance to the Sun (Astronomical Unit) as 140,000,000 km.
Current value is 8.8 arcseconds, or 149,597,892 km.
B
D
B
D
A
Parallax angle A
1 arcsecond = (1 degree)/3600
The angle between these two lines is 2.5 degrees
The angle of 9 arcseconds is 1000 times smaller!
One needs baselines larger than the size of the Earth!
Aristarchus of Samos (~310-230 BC)
Using the distance between the Earth and the Moon as a baseline
Using the Earth’s orbit as a baseline
Even for the nearest stars the parallax is 0.3 second of arc.
This is 10,000 times smaller than 1 degree!
Aristotle (384-322 BC) could not find measurable parallax, so
he concluded the Earth was not moving
Retrograde
(westward)
motion of a
planet occurs
when the
Earth passes
the planet.
The Copernican Revolution
Nicolaus Copernicus (1473 – 1543):
Heliocentric Universe (Sun in the Center)
Born: 19 Feb 1473 in Torun, Poland
Died: 24 May 1543 in Frombork, Poland
Church cleric, but rejected a 2000-yr old paradigm
Seven axioms written in a pamphlet “Little Commentary” (1514)
1. There is no one centre in the universe.
2. The Earth's centre is not the centre of the universe.
3. The centre of the universe is near the sun.
4. The distance from the Earth to the sun is imperceptible compared with the
distance to the stars.
5. The rotation of the Earth accounts for the apparent daily rotation of the stars.
6. The apparent annual cycle of movements of the sun is caused by the Earth
revolving round it.
7. The apparent retrograde motion of the planets is caused by the motion of the
Earth from which one observes.
Johannes Kepler (1571 – 1630)
Kepler’s Laws of Planetary Motion
1. The orbits of the planets are ellipses with the
sun at one focus.
c
Eccentricity e = c/a
A New Era of Science

I. If F = 0, velocity v  const


Change of motion = acceleration a  v / t
II.
 
ma  F
A ball experiences acceleration  there must be a force!
Acceleration (a) is the change of a
body’s velocity (v) with time (t):
a
a = v/t
Velocity and acceleration are directed
quantities (vectors)!
v
Different cases of acceleration:
1. Acceleration in the conventional
sense (i.e. increasing speed)
2. Deceleration (i.e. decreasing speed)
3. Change of the direction of motion
(e.g., in circular motion)
Curved path of the Moon  there must be some force!
Newton’s law of gravitation
Newton’s Cannon
Measurements of parallaxes of the Sun, Moon, planets and stars
seemed to fit Newton’s theory amazingly well …
Clockwork universe
One little speck on the brilliant face of Newton’s theory:
The advance of the perihelion of Mercury
Mercury: the closest planet to the
Sun
Perihelion = position
closest to the sun
Mercury
Sun
Perihelion: 46 million km; Aphelion: 70 million km
Aphelion
= position
furthest
away
from the
sun
In reality the orbits deviate from elliptical:
Mercury's perihelion precession: 5600.73 arcseconds per century
Newtonian perturbations from other planets: 5557.62 arcseconds per century
Remains unexplained: 43 arcseconds/century (Le Verrier 1855)
Predicted the presence and position of Neptune
from irregularities in Uranus’s orbit
Neptune was found in 1846 exactly at the
predicted position
Urbain Le Verrier 1811-1877
In the eyes of all impartial men, this discovery [Neptune] will remain
one of the most magnificent triumphs of theoretical astronomy …
Arago
I do not know whether M. Le Verrier is actually the most detestable man
in France, but I am quite certain that he is the most detested.
A contemporary
In 1855 Le Verrier found that the perihelion of Mercury advanced
slightly more than the Newtonian theory predicted.
He and others tried to explain it with a new planet Vulcan, new
asteroid belt, etc.
By the beginning of the XX century, it became clear that
Newtonian gravity has other problems
Problem with Action at a Distance

 Gm1m2 
F1   F2 
r0
2
r
m1

F1

F2
m2
If ball 1 moves, ball 2 instantaneously feels it.
Direct, instantaneous connection between
cause and effect!
Faster than light propagation??
Einstein’s idea:
Newton’s theory is a weak-gravity limit of a more
general theory: General Relativity
Even in the weak gravity of the Earth and the Sun,
there are measurable deviations from Newtonian
mechanics and gravitation law!
• Precession of Mercury’s perihelion
• Bending of light by the Sun’s gravity
General Relativity predicts new effects, completely absent in the
Newton’s theory: black holes, event horizon, gravitational waves.
Gravity is a strange force. It has a unique property:
All bodies in the same point in space experience the same acceleration!
m
mM
F G 2
R
R
M
F
M
a
G 2
m
R
Acceleration of Gravity
Iron ball
Wood ball
Acceleration
of gravity is
independent
of the mass of
the falling
object!
This means that in the freely-falling elevator cabin you
don’t feel any effects of gravity! You and all objects
around you experience the same acceleration.
In outer space you can imitate the effect of gravity by acceleration.
"You mighta seen a house fly, maybe even a superfly, but you ain't
never seen a donkey fly!"
Donkey
Equivalence Principle
In 1907, Einstein was preparing a review of special relativity when he
suddenly wondered how Newtonian gravitation would have to be
modified to fit in with special relativity. At this point there occurred to
Einstein, described by him as the happiest thought of my life , namely
that an observer who is falling from the roof of a house experiences no
gravitational field. He proposed the Equivalence Principle as a
consequence:... we shall therefore assume the complete physical equivalence of a
gravitational field and the corresponding acceleration of the reference
frame. This assumption extends the principle of relativity to the case of
uniformly accelerated motion of the reference frame.
Immediate consequences of the Equivalence Principle:
• Bending of light in the gravitational field
• Time flow and frequency of light are changed in the
gravitational field
The bulb emits flashes of light 2 times per second
c
Acceleration a = gravity g
An observer on the floor receives flashes faster than 2 times per second
First observed on the Earth by Pound and Rebka 1960:
relative frequency shift of 10-15 over the height of 22 m.
Light should be bent in the gravitational field
If gravity can be eliminated by motion, no special force of gravity
is needed!
The force of gravity is actually the acceleration you feel when
you move through space-time
How to explain that in the absence of any force the
trajectories are not straight lines?
Because space and time are curved!
All bodies experience the same acceleration, but
only in a small region of space. In another region
this acceleration is different. Time flows with a
different rate, and paths are bent differently in these
two regions.
m
M
a1  G 2
R1
M
a2  G 2
R2
R1
M
R2
Main idea:
Space-time gets curved by masses. Objects traveling in curved spacetime have their paths deflected, as if a force has acted on them.
“Curvature” of time means that the time flows with a different rate
in different points in space
"Matter tells spacetime how to bend and spacetime returns the
complement by telling matter how to move."
John Wheeler
Shortest paths are called geodesics; they are not straight lines!
The shortest path between two cities is not a straight line
Low density star
High density star
About 1912 Einstein realized that the geometry of our world
should be non-Euclidean.
He consulted his friend Grossmann who was able to tell Einstein of the
important developments of Riemann, Ricci and Levi-Civita.
G.F.B. Riemann
(1826-1866)
When Planck visited Einstein in 1913 and Einstein told him the present
state of his theories Planck said:
As an older friend I must advise you against it for in the first place you
will not succeed, and even if you succeed no one will believe you.
Several versions of Einstein’s GR in 1913-1914 were wrong.
Only in November 1915, after correspondence with Levi-Civita and
Hilbert, Einstein published a paper with correct equations.
Hilbert also published correct equations, in fact 5 days earlier than
Einstein.
On the 18th November Einstein made a discovery about which he wrote
For a few days I was beside myself with joyous excitement . He
explained the advance of the perihelion of Mercury with his theory.
Bending of light
Two British expeditions in 1919 confirmed
Einstein’s prediction.
The shift was about 2 seconds of arc, as predicted
Gravitational lensing
Gallery of lenses (Hubble Space Telescope)
Black holes
A black hole is a region of space-time from which
nothing can escape, even light (first suggested by
Laplace in 1796).
Schwarzschild radius
1
GMm
2GM
2
mc 
 Rs  2
2
Rs
c
Derivation is wrong and picture is
wrong, but the result is correct
To make a black hole from a body of mass M, one
needs to squeeze it below its Schwarzschild’s radius
2GM
Rs  2
c
Rs
K. Schwarzschild
Schwarzschild radius: event horizon for a nonrotating body
No signals can reach an outside observer from inside the
event horizon! This is a point-of-no-return for everything that
crosses it.
The curvature of a 2D slice of a spherically symmetric black hole
Curvature becomes infinite as we approach the singularity r =0
Gravitational bending of light paths around a black hole
Approaching a black hole
Circling around a black hole
Time dilatation
Rs  2

ds   1  dt
r 

2
t o
t 
Rs
1
r
t0
t
As measured by a distant observer, clocks slow down
when approaching a massive object
Tidal forces and contraction of space squeeze and
stretch the astronaut. Lateral pressure is 100 atm at a
distance of 100 Rs from the event horizon
Black holes are NOT big cosmic bathtub drains!
Approaching a black hole R ~ Rs
(strong field): gravity pull runs away
Far from a black hole R >> Rs
(weak field): Newtonian gravity
law holds
GM
ag 
R
2
R
1 s
R
GM
ag  2
R
If our Sun collapses into a black hole, we won’t
see any difference in the gravitational pull (but it
will be VERY cold)
Black holes in the Universe
1. Formation of galaxies
2. Collapse of massive stars
3. Early Universe?
How to find the object that does not emit any radiation?
By its effect on nearby objects
The Galactic Center
Our view (in visible light) towards the galactic center (GC)
is heavily obscured by gas and dust
Extinction by 30 magnitudes
 Only 1 out of 1012 optical photons makes its
way from the GC towards Earth!
Galactic center
Wide-angle optical view of the GC region
If one looks at this region with big telescopes and nearinfrared cameras one can see lots of stars. If one takes
pictures every year it seems that some stars are moving
very fast (up to 1500 kilometers per second). The fastest
stars are in the very center - the position marked by the
radio nucleus Sagittarius A* (cross).
Distance between stars
is less that 0.01 pc
A Black Hole at the Center of Our
Galaxy
By following the orbits of individual stars near the
center of the Milky Way, the mass of the central black
hole could be determined to ~ 2.6 million solar masses
Black hole vicinity is probably very messy …
Cores of other galaxies show an accretion disk with
a possible black hole
“All hope abandon, ye who enter here”
Dante
Second scenario of black-hole formation:
Death of massive stars
Gravitational collapse of the iron core
Supernova explosion
Mass transfer in a binary system
Binary systems
3
If we can calculate the total mass and
measure the mass of a normal star
independently, we can find the mass of
an unseen companion
a
M1  M 2  2 ;
P
a – in AU
P – in years
M1+M2 – in solar masses
Still, this is an indirect evidence. Proving the existence of
black holes remains one of the greatest problems in physics.
1912
You cannot tell the difference
between acceleration and
gravity!