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Дата изменения: Thu Oct 7 22:32:21 1999 Дата индексирования: Tue Oct 2 03:35:27 2012 Кодировка: Поисковые слова: rho ophiuchi |
Cosmology and the Origin of Life
We can derive Kepler's Third Law from Newton's equations as follows:
Step 1:
Assume that M1 is a small mass in a circular
orbit about a much larger mass M2 . We can write down the Force
law on M1 using Newton's formulations:
![]() ![]()
Step 2:
Combining terms yields:
Step 3:
In an orbit governed by a central force, the
centripetal acceleration, a is given by:
Step 4: For a circular orbit, the circular velocity, Vc is the total distance traveled (the circumference of the circle) divided by the orbital period, P or Vc = 2p R/P which then yields: ![]() |
The point of
this derivation was to show that only the R-2 force law
can yield Kepler's Third Law. No other force law will work. If this
gravitional force law is universal then Kepler's laws must also
be universal.
This was followed in 1839 by Thomas Henderson's measurement of about 1 arc-second for Alpha Centauri and the 1840 measurement of Alpha Lyrae (Vega) at 0.26 arcseconds.
1 parsec is the distance a star would have to be at to have a parallactic angle of 1 arcsecond. This distance is equivalent to 3.26 light years.
By 1840, it became clear that even the closest stars were more than one million times farther away than our Sun.
The Universe was now a large place and the intrinsic energy output of the stars had to be huge in order that their light could reach us.
Now back to , who makes two mistakes:
Einstein Rescues Us:
provides absolute frame and finite communication
time. If not, you could have this .
Okay so we live in curved spacetime and now you are telling me that the Universe is exanding:
Uniform Expansion of the Universe:
Hubble noticed a correlation between recessional velocity and distance. This is known as the Hubble law:
where V is velocity (in km/s), D is distance (in megaparsecs), H is the Hubble constant (present day expansion rate of the Universe)
![]() |
The line through the data is a "best fit" linear relationship which shows that there is a linear relationship between the the velocity at which a galaxy moves away from us and its distance. This linear relatinship is consistent with a model of uniform expansion for the Universe. |
This simple relation implies something remarkable about the Universe.
At some earlier time, all the galaxies had to have been together
in the same space at the same time
the Universe was once
really small. It is important to realize that the galaxies are
stuck to the surface of the universe by gravity and its the surface
that expands. The galaxies themselves are not moving but travel
along with the surface as shown here .
So what do we know now:
Photons have an energy related mass. Photons are therefore effected
by gravity. Light is bent in a strong gravitational field as the
surface of the universe goes from being flat (light travels in
a straight line) to curved (light follow the curved trajectory).
This principle is shown in this
animation .
The distribution of mass on in the universe determines the detailed
shape of the surface light is constrained to follow this surface
and this allows the universe to be observed!
Returning now to the expansion of the Universe with the help of some handy demonstration ants:
Horizons and the Expansion Age of the Universe:
V = HD c = HD ==> D =c/H ==> This is our causal horizon -
beyond this distance something would have to travel faster than the
speed of light in order to communicate with us. All observers
are surrounded by such a horizon.
Horizons are okay. Our assumption about homogeneity means that the stuff beyond the horizon is the same stuff we already know about. This assumption must be correct due to horizon overlaps and causality.
Back to the Ants glued to the balloon:
Example: I attach the balloon to a slow pump which increases the
radius of the balloon by one foot each day. This is the expansion
rate that I measure. I measure the balloon to have a radius of
8 feet. This means the expansion age of the balloon is 8 days.
V = Hd ==> 1/H = D/V
Distance/Velocity = Time
1/H = the expansion age of the Universe.
This is how long the
Universe has been expanding. What it was doing prior to the expansion
is anybody's guess.
If you know the rate of inflation of the balloon (the expansion
rate of the surface) and the surface area of the balloon (which
is proportional to its radius) then you can determine how long it
has taken for the balloon to reach its present size.