Download Lecture 4 - The Local Group

Lasers, GPS, AO, &
spy satellites…
[as time permits]
[Oct 4, 2016]
As with all course material (including homework, exams),
these lecture notes are not be reproduced, redistributed,
or sold in any form. Announcements
Homework #1 on class website (due Today, Oct 4th by 5pm)
A Brief Aside…
In 2015, NASA released a large collection of
images from the Apollo Program.
[>8000 images]
https://www.flickr.com/photos/projectapolloarchive/albums
Apollo 9
Apollo 11
Apollo 12
Any idea
what this is?
Any idea
what this is?
a reflector
plate left
behind by
Apollo 11,
14, or 15.
Part of the
lunar laser
ranging
experiment
What is light?
• Electromagnetic Radiation
–what we call “light” is a type of electromagnetic radiation that we can see (380nm-740nm)
• Light carries energy and can transmit information.
• It moves very quickly, but is not infinitely fast.
Distance from the Earth to the Moon in 1.28 seconds
That’s 384,000 km in 1.28 seconds
Speed of light c ~ 300,000 km/s
What is light?
• Electromagnetic Radiation
–what we call “light” is a type of electromagnetic radiation that we can see (380nm-740nm)
• Light carries energy and can transmit information.
• It moves very quickly, but is not infinitely fast.
Distance from the Earth to the Moon in 1.28 seconds
That’s 384,000 km in 1.28 seconds
Speed of light c ~ 300,000 km/s
What is light?
• Electromagnetic Radiation
–what we call “light” is a type of electromagnetic radiation that we can see (380nm-740nm)
• Light carries energy and can transmit information.
• It moves very quickly, but is not infinitely fast.
Distance from the Earth to the Moon in 1.28 seconds
That’s 384,000 km in 1.28 seconds
Speed of light c ~ 300,000 km/s
What is light?
• Gamma-rays, x-rays, ultraviolet, visible, infra-red, TV, and radio are all light.
• What’s the difference?
What is light?
• Gamma-rays, x-rays, ultraviolet, visible, infra-red, TV, and radio are all light.
• What’s the difference? -- wavelength
The Electromagnetic
Spectrum
The Electromagnetic
Spectrum
ultraviolet infrared
What is a laser?
What is a laser?
laser = light amplification by stimulated emission of radiation
What is a laser?
laser = light amplification by stimulated emission of radiation
http://spaceplace.nasa.gov/laser/en/
Measuring the distance to the Moon with Lasers
Earth
Moon
Measuring the distance to the Moon with Lasers
t ~ 2.5 seconds
Earth
Moon
Measuring the distance to the Moon with Lasers
t ~ 2.5 seconds
Earth
Moon
speed of light (300,000 km/s)
d=cxt/2
distance to moon
[distance measured to ~2 cm precision]
time for laser pulse to reflect back
Lick Observatory — Mt Hamilton (San Jose, CA)
Why do you need a telescope for lunar ranging…?
Where we left off from Lec 3…
What’s that big ring of satellites at large distances?
Note that this is a ring…an equatorial ring of satellites!
Recall that all objects in orbit rotate about the Earth’s center
of mass (i.e. its center assuming it is symmetric).
There are many
geosynchronous orbits, but
only one geostationary orbit.
orbits in 24 hours
orbits over a fixed
spot on Earth
What other satellites do you want to put
in geostationary orbit?
What other satellites do you want to put
in geostationary orbit?
You might think that GPS (Global
Positioning System) satellites are in
geostationary orbit, but they are not.
Instead, they are located at an altitude of ~20,000 km
(vs ~35,600 km for geostationary).
How quickly are GPS satellites moving?
√
Vorbit =
G Mearth
d
How quickly are GPS satellites moving?
√
Vorbit =
G Mearth
d
d = 20,000 km
G = 6.67 x 10-11 m3 kg-1 s-2
Mearth = 6 x 1024 kg
How quickly are GPS satellites moving?
√
Vorbit =
G Mearth
d
d = 20,000 km
+ 6400 km
G = 6.67 x 10-11 m3 kg-1 s-2
Mearth = 6 x 1024 kg
How quickly are GPS satellites moving?
√
Vorbit =
G Mearth
d
Vorbit ~ 4 km/s
d = 20,000 km
+ 6400 km
G = 6.67 x 10-11 m3 kg-1 s-2
Mearth = 6 x 1024 kg
So what is the period of their orbit?
P = 2 𝜋 R / Vorbit
Vorbit ~ 4 km/s
R = 26,400 km
So what is the period of their orbit?
P = 2 𝜋 R / Vorbit
P = 42,000 seconds
P = 12 hours
Vorbit ~ 4 km/s
R = 26,400 km
So what is the period of their orbit?
P = 2 𝜋 R / Vorbit
Vorbit ~ 4 km/s
R = 26,400 km
P = 42,000 seconds
P = 12 hours
Thus, GPS satellites orbit the Earth twice per day.
Ok, so how does GPS work?
Ok, so how does GPS work?
To understand that, we have to step back and discuss
the work of one of the most famous physicists ever?
Albert Einstein
(1879 - 1955)
1905
No one before or since has widened the horizons of physics in so short a
time as Einstein did in 1905. -Pais
In 1905 Einstein (age 26) produced six pieces of scientific
work. Three papers (including his PhD thesis) would help
solidify the existence of molecules & “kinetic theory”.
Three others would usher in the era of Modern Physics:
1. March: The light-quantum hypothesis
• Says that light “hits like a particle”
• Will win Nobel Prize for this in 1921
• Fundamental in development of Quantum Mechanics
2. June: First paper on Special Relativity
• Length contraction
• Time dilation
3. September: Second paper on Special Relativity
• E = mc2
Special Relativity
Einstein’s Assumptions:
1) The laws of physics take the same form in all
uniformly moving reference frames.
2) In any uniformly moving reference frame, the
velocity of light (c) is the same, whether emitted
by a body at rest or a body in uniform motion.
Time Dilation
Moving clocks seem to run slowly
when viewed by people who are
not moving with the clocks.
This is not an artifact of the clocks,
time itself does not “tick by” at
the same rate. It looks to a
stationary observer that time ticks
by more slowly on moving rockets
than it does to a person riding
along on the rocket!
Moving train. Bulb & mirror. Light takes
time to bounce back from the mirror
v
h
h
Non-moving observer: the
distance the light travels is
longer, and therefore it must
take longer for light to complete
its path.
This result is because the speed of light is
not affected by the speed of the train!
v
v
v
v
distance traveled by light = 2 x d
d
{
v
d
{
distance traveled by light = 2 x d
d
{
v
d
2 x √ x2 + d2
{
d
x
distance traveled by light is longer
{
distance traveled by light = 2 x d
d
speed = distance
/ time
v
{
d
special relativity says that the
speed of light is the same for
all observers.
{
thus, since the distance
traveled is longer for the
2 + d2
x
√
stationary observer, the time
must also be longer.
d
x
distance traveled by light is longer
{
Time Dilation
Moving clocks seem to run slowly
when viewed by people who are
not moving with the clocks.
This is not an artifact of the clocks,
time itself does not “tick by” at
the same rate. It looks to a
stationary observer that time ticks
by more slowly on moving rockets
than it does to a person riding
along on the rocket!
Moving with train:
Measures 2 seconds.
Watching
the train:
Measures
2.5 sec.
Time Dilation
Moving clocks seem to run slowly
when viewed by people who are
not moving with the clocks.
This is not an artifact of the clocks,
time itself does not “tick by” at
the same rate. It looks to a
stationary observer that time ticks
by more slowly on moving rockets
than it does to a person riding
along on the rocket!
Moving with train:
Measures 2 seconds.
Watching
the train:
Measures
2.5 sec.
1907:
“The happiest thought of my life…”
• While others were still thinking about the implications of special relativity,
Einstein was already moving forward.
• Einstein realized that he would also have to modify Newton’s gravity in order
for it to sync it with Special Relativity:
“Then there occurred to me the happiest thought of my life:
The gravitational field has only a relative existence… Because for an
observer falling freely from the roof of a house there exists -- at least in his
immediate surroundings -- no gravitational field.”
Towards his greatest work, General Relativity…
1907: The Equivalence Principle: gravity and acceleration are equivalent.
It occurred to Einstein – that if a man were falling from the roof of his
house and tried to let anything drop, it would only move alongside him,
thus indicating the equivalence of acceleration and gravity.
1911: Einstein realized that the bending of light in a gravitational field,
which he knew in 1907 was a consequence of the equivalence principle,
could be checked with astronomical observations.
Towards his greatest work,
General Relativity…
I have just completed the most splendid work of my life...
--to his son Hans Albert, 1915
1912-1915: Hunting for The Theory. Einstein began to search for particular
equations -- ones that would extend relativity to objects moving in an arbitrary
way with respect to one another. The search was arduous with entire years spent in
blind alleys. Einstein had to master more elaborate mathematical techniques than
he had ever expected to need...
Planck: “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.”
1915: Success. Albert Einstein completes his Theory of General Relativity
Gravity <=> curved space
General Relativity
The theory appeared to me then, and still does, the
greatest feat of human thinking about nature, the most
amazing combination of philosophical penetration,
physical intuition, and mathematical skill.—Max Born
Einstein’s equations:
Matter tells space how to curve
and space tells matter how to
move
No need for “Force” of Gravity! All motion is along “straight lines”
in curved space-time.
1915: General theory of relativity
Gravity curves space
Predictions:
1. Starlight will bend around the Sun
2. Mercury's orbit will precess at a different rate
than predicted by Newton.
Testing Relativity
Einstein published a paper calling on astronomers to test two
predictions of his theory:
• Light from a distant star should bend if it crosses
close to the gravitational field of the Sun. This
should be measurable during a solar eclipse.
• Solar spectral lines should “redshift” relative to
spectral lines produced on Earth's surface
because of the gravitational field.
1919: The bending of starlight
Starlight passing near the sun should be slightly deflected by
gravity. This deflection could be measured when the Sun's own
light was blocked during an eclipse. Einstein predicted a specific
amount of deflection.
1919 : Arthur Eddington leads a solar eclipse expedition which
detected gravitational deflection of light by the Sun
Announcement of the eclipse results caused a
sensation. This proved Einstein’s prediction,
lending validity to his theory of General
Relativity. Einstein received world fame.
Bending of starlight
When asked what he would do if the
measurements of bending starlight at
the 1919 eclipse contradicted his
general theory of relativity, Einstein
said,
"Then I would feel sorry for the good
Lord. The theory is correct."
New York Times on November 10, 1919.
What does that have to do with GPS?
➡GPS
• 24
would not work without Special Relativity/GR.
satellites orbiting at ~20k km
(i.e. orbiting at ~14k km/hr)
• ≥ 4 satellites always visible
• satellites carry atomic clocks (accurate to ~1 nanosec)
• satellites transmit to receivers on
Earth via radio waves.
How GPS works in 2-d...
How GPS works in 2-d...
How GPS works in 2-d...
Measuring the distance to...
1 satellite fixes our location to a sphere:
2 satellites fixes our location to a circle:
3 satellites fixes our location to 2 points:
(1 of which is on the surface of the Earth)
Measuring the distance to...
1 satellite fixes our location to a sphere:
2 satellites fixes our location to a circle:
3 satellites fixes our location to 2 points:
(1 of which is on the surface of the Earth)
a 4th satellite allows GPS to reliably determine altitude.
Why GPS fails without relativistic physics…
To measure location to a precision of ~15m, we need to be able
to measure time to ~50 nanoseconds (GPS does better than this). Due to relativistic effects, the clocks on the GPS satellites tick
~40 microseconds faster per day (recall that GPS satellites are
orbiting at ~14,000 km/hr and farther from Earth). That seems like a small time difference: 40 μs - 50 ns = 40 x 10-6 s - 50 x 10-9 s ~ 40 x 10-6 s
[this is like $40,000 minus $50 = $39,950.]
But light moves at 300,000 km/s…so this translates to a distance of 300,000 km/s x 40 μs = 12 km (per day!!)
Without relativity, GPS would be useless within a few minutes.
Back to Lunar
Ranging…
Measuring the distance to the Moon with Lasers
Earth
Moon
Measuring the distance to the Moon with Lasers
t ~ 2.5 seconds
Earth
Moon
Measuring the distance to the Moon with Lasers
t ~ 2.5 seconds
Earth
Moon
speed of light (300,000 km/s)
d=cxt/2
distance to moon
[distance measured to ~2 cm precision]
time for laser pulse to reflect back
Measuring the distance to the Moon with Lasers
t ~ 2.5 seconds
Earth
Moon
speed of light (300,000 km/s)
d=cxt/2
distance to moon
[distance measured to ~2 cm precision]
time for laser pulse to reflect back
Why is the calculation more difficult than this in detail?
Why do we care? — test of GR!!
See you Thursday…