ASTR1010
- INTRODUCTION TO ASTRONOMY
The Solar System
February 20,
2009
As
the 19th century dawned, it seemed that NewtonÕs Theory of
Gravitation had won the day. All
sorts of motion could be described by the equations that we have looked at (all
based on F = ma and F = GM1M2/d2 ). However, it soon became clear that the planet Mercury was
not obeying NewtonÕs Laws. At this
point, scientists can do one of two things: either reject the theory and
replace it with a better one, or find some subtlety that had been missed in
examining the problem the first time.
Astronomers at first chose the second path and spent a good deal of time
during the 19th century looking for a planet interior to MercuryÕs
orbit. This planet would disturb
Mercury in just the right way to explain the observed discrepancy in the precession of the perihelion of MercuryÕs
orbit. Because of the
gravitational effect of other bodies in the Solar System (i.e., Venus, Earth,
Jupiter) the orbit of Mercury is not a closed ellipse (the gravitational
attraction from the other planets slightly disturb the gravitational interaction
between the Sun
and Mercury). Instead, the
perihelion (or the semimajor axis, or the major axis
depending on how you want to look at it) slowly turns or precesses
in the Mercury/Sun orbital plane.
This rate of precession is very slow and NewtonÕs Laws predicted that it
should occur at the rate of 5557 seconds of arc per century. However, observations showed that the
perihelion was precessing at a rate of 5600 seconds
of arc per century, a discrepancy of 43 seconds of arc every 100 years (such a small
angular shift that over the course of a year or two you couldnÕt even measure
the effect). With decades of
observation, it became clear that the discrepancy was real, so, as mentioned
above there were only two choices for astronomers: either there was a problem
with NewtonÕs Laws, or there was an undiscovered planet orbiting very close to
the Sun and disturbing MercuryÕs orbital motion in just the right way to
produce the discrepancy. By the
end of the 19th century, it had become clear that such a planet does
not exist. Thus, it turned out we
had to Òfix upÓ NewtonÕs Laws.
This was done at the dawn of the 20th
century by Albert Einstein who came up with a new way of looking at gravity and
orbital motion. He hadnÕt set out
originally to explain the precession of MercuryÕs orbit; instead, he was
looking for a way to make the laws of electromagnetism the same for all
observers independent of their state of motion. The speed of light, c, enters naturally into the equations
of electromagnetism (the so-called Maxwell equations) and this causes a problem
between people doing experiments in stationary as opposed to moving coordinate
systems (if youÕre in a laboratory moving at some velocity, v, then what is the speed of light, is
it the same value as for the stationary observer or is it c plus or minus v?).
Worrying about and trying to resolve this problem led Einstein to develop his Special Theory of Relativity.
Einstein
began with two postulates (statements which are given or assumed but cannot be
proven):
1) All observers should arrive at
the same physical phenomena regardless of their velocity. In other words, the laws of physics
shouldnÕt depend on whether you are moving at constant velocity or not.
2) The propagation of signals occurs
at a finite speed, and this speed is that of light.
These harmless-sounding postulates led to a revolution in thought. Using these postulates, Einstein was led to some startling conclusions when examining what we mean by a simultaneous event. Basically, two observers in relative motion with respect to one another cannot agree on whether two events are simultaneous or not. This is due to the finite speed of light. If light could travel infinitely fast, then there would be no problem; but instead, the second postulate forces you into this non-intuitive situation.
Now, if two observers cannot judge with absolute certainty whether two events are simultaneous, then there seems to be an issue with time. If time was absolute for all observers (like Newton thought), then there should be no problems judging the simultaneity of events. It turns out that with the second postulate, the only way to reconcile events from the perspective of two observers which are moving with respect to each other is to conclude that time is flowing differently for each observer. This effect is called time dilation and can be stated as follows: if you are looking at a moving clock, you will see it ÒtickingÓ (marking time) more slowly than your clock (which is attached to you in your frame of reference and is not moving with respect to you). The formula for time dilation is:
t(moving) = t(observer) (1
– v2/c2)½ .
There
are other effects besides time dilation that are forced on us by the two
postulates above. These effects
include the contraction of a moving object in the direction of motion and the
increase of mass for a moving object.
To illustrate length contraction, letÕs say that you measure something
that is not moving with respect to you (letÕs say a bar of aluminum) and you
determine it has a certain length, Lo. Now, you observe the bar moving in some direction along its
length. What you measure for its
length while itÕs moving (call this length L(moving))
is:
L(moving) = Lo (1
– v2/c2)½ .
Because
the factor (1 – v2/c2)½ is always equal to or less than 1
(because v has to be less than c), L(moving) will always be smaller than Lo
(they are equal if v = 0; in other words, if the bar is not moving with respect
to you). This observation is an
inescapable consequence of the two postulates above. We will see in the next lecture that the real problem is
that the two postulates create a world where it is difficult to compare what is
happening between moving observers.
They have a hard time measuring the same things so no wonder they donÕt
get the same answers. The
observers think they are measuring the same things, but they turn out to be
wrong even when their intuition tells them they are measuring the same
events. More on
this later.
The
last effect that the two postulates force on us is that when you observe the
mass of a moving object, it is larger than when you determine its mass while it
is standing still. The formula is
similar to the time dilation formula:
m(moving) = m(rest) / (1 – v2/c2)½ .
It
is from this relation that the famous E = mc2 formula comes from.
So
the two postulates of Special Relativity lead us into a strange world where we
observe moving clocks to tick more slowly than our clock, where moving objects
look like they shrink in their direction of motion, where we measure larger
masses for moving objects, and where determining whether events are
simultaneous or not depends on the relative motion of the observers. The question is, is this crazy world
really our world? In other words,
do experiments bear out these effects?
The
answer is yes. All experiments
done to date confirm that these effects really work out as the equations of
Special Relativity dictate.
Elementary particles can be speeded up to velocities close to the speed
of light and the so-called Òrelativistic effectsÓ described above are indeed
observed in particle accelerators in laboratories on a daily basis. Atomic clocks put on airplanes and
flown for hours on end come back having Òticked lessÓ than their stationary
counterparts on the ground. We
donÕt notice relativistic effects because the factor (1 – v2/c2)½ doesnÕt amount to much unless you
are going a sizeable fraction of the speed of light. Thus, our common sense intuition is based on velocities
where the factor (1 – v2/c2)½ is basically equal to 1 (see
problem 1 below). No wonder we are
not normally aware of these weird phenomena!
Problems:
1)
Calculate the value of (1 – v2/c2)½ when you are going 10 meters/second,
1000 meters/second, 1 million meters/second.
2)
An object is moving at 6/10 of the speed of light. If your clock ticks off 100 seconds,
how many seconds have ticked off on the objectÕs clock? Repeat the calculation for an object
moving at 0.99c.
3)
LetÕs say that a bar of aluminum which is 10.0 centimeters long is
moving at 10 meters/second with respect to you. How long does it look to you? How long does it look if it is moving 1 million meters per
second?
4)
State what the postulates of Special Relativity are in your own
words. Why is the second postulate
so odd?
5)
A proton has a mass of 1.67 x 10-27 kg when it is at rest. What is its mass when it is going at
0.6c? What is its mass when it is
going at 0.99c? What is its mass
when it is going at 0.9999c?