1 leg man said 10 miles a day
as of feb 26/2005 mine is 10.14+ a day
today feb 27 2005 I still here.
proff as the world turns.
So as I have 2 legs and most here in rr do.
I can say at warm up time still:
10.14 miles × 57 days is well a head in race of 1 leg man at 10 miles
a day saids:
proff. if interested=
I post. most in michigan do not.
time to ride soon. KYA
like being :Haveing your ass kick in miles by a one legged man is too
say.
How can you look at your self and be with out shame.........
Kepler's laws of planetary motion
From Wikipedia, the free encyclopedia.
Johannes Kepler's primary contributions to astronomy/astrophysics
were the three laws of planetary motion. Kepler derived these laws, in
part, by studying the observations of Brahe. Isaac Newton would later
design his laws of motion and universal gravitation and verify that
Kepler's laws could be derived from them. The generic term for an
orbiting object is "satellite".
Contents [showhide]
1 Kepler's laws of planetary motion
2 Kepler's first law
2.1 Connection with Newton's laws
3 Kepler's second law
3.1 Proof of Kepler's second law:
4 Kepler's third law (harmonic law)
5 Applicability
6 Application
7 Kepler's understanding of the laws
8 See also
[edit]
Kepler's laws of planetary motion
Kepler's first law (1609): The orbit of a planet about a star is an
ellipse with the star at one focus.
Kepler's second law (1609): A line joining a planet and its star sweeps
out equal areas during equal intervals of time.
Kepler's third law (1618): The square of the sidereal period of an
orbiting planet is directly proportional to the cube of the orbit's
semimajor axis.
[edit]
Kepler's first law
The orbit of a planet about a star is an ellipse with the star at one
focus.
There is no object at the other focus of a planet's orbit. The semimajor
axis, a, is the average distance between the planet and its star.
[edit]
Connection with Newton's laws
Newton proposed that "every object in the universe attracts every other
object along a line of the centers of the objects proportional to each
objects mass, and inversely proportional to the square of the distance
between the objects."
This section proves that Kepler's first law is consistent with Newton's
laws of motion. We begin with Newton's law F=ma:
Here we express F as the product of its magnitude and its direction.
Recall that in polar coordinates:
In component form we have:
Now consider the angular momentum:
So:
where is the angular momentum per unit mass. Now we substitute. Let:
The equation of motion in the direction becomes:
Newton's law of gravitation states that the central force is inversely
proportional to the square of the distance so we have:
where k is our proportionality constant.
This differential equation has the general solution:
Replacing u with r and letting ?0=0:
..
This is indeed the equation of a conic section with the origin at one
focus. Q.E.D.
[edit]
Kepler's second law
A line joining a planet and its star sweeps out equal areas during equal
intervals of time.
This is also known as the law of equal areas. Suppose a planet takes 1
day to travel from points A to B. During this time, an imaginary line,
from the Sun to the planet, will sweep out a roughly triangular area.
This same amount of area will be swept every day.
As a planet travels in its elliptical orbit, its distance from the Sun
will vary. As an equal area is swept during any period of time and since
the distance from a planet to its orbiting star varies, one can conclude
that in order for the area being swept to remain constant, a planet must
vary in velocity. Planets move fastest when at perihelion and slowest
when at aphelion.
This law was developed, in part, from the observations of Brahe that
indicated that the velocity of planets was not constant.
This law corresponds to the angular momentum conservation law in the
given situation.
[edit]
Proof of Kepler's second law:
Assuming Newton's laws of motion, we can show that Kepler's second law
is consistent. By definition, the angular momentum of a point mass with
mass m and velocity is :
..
where is the position vector of the particle.
Since , we have:
taking the time derivative of both sides:
since the cross product of parallel vectors is 0. We can now say that
is constant.
The area swept out by the line joining the planet and the sun, is half
the area of the parallelogram formed by and .
Since is constant, the area swept out by is also constant. Q.E.D.
[edit]
Kepler's third law (harmonic law)
The square of the sidereal period of an orbiting planet is directly
proportional to the cube of the orbit's semimajor axis.
P2 ~ a3
P = object's sidereal period in years
a = object's semimajor axis, in AU
Thus, not only does the length of the orbit increase with distance, also
the orbital speed decreases, so that the increase of the sidereal period
is more than proportional.
See the actual figures: attributes of major planets.
Newton would modify this third law, noting that the period is also
affected by the orbiting body's mass, however typically the central body
is so much more massive that the orbiting body's mass may be ignored.
(See below.)
[edit]
Applicability
The laws are applicable whenever a comparatively light object revolves
around a much heavier one because of gravitational attraction. It is
assumed that the gravitational effect of the lighter object on the
heavier one is negligible. An example is the case of a satellite
revolving around Earth.
[edit]
Application
Assume an orbit with semimajor axis a, semiminor axis b, and
eccentricity ?. To convert the laws into predictions, Kepler began by
adding the orbit's auxiliary circle (that with the major axis as a
diameter) and defined these points:
c center of auxiliary circle and ellipse
s sun (at one focus of ellipse);
p the planet
z perihelion
x is the projection of the planet to the auxiliary circle; then
y is a point on the circle such that
and three angles measured from perihelion:
true anomaly , the planet as seen from the sun
eccentric anomaly , x as seen from the centre
mean anomaly , y as seen from the centre
Then
area cxz = area cxs + area sxz = area cxs + area cyz
giving Kepler's equation
..
To connect E and T, assume r = length sp then
and rsinT = bsinE
which is ambiguous but useable. A better form follows by some trickery
with trigonometric identities:
(So far only laws of geometry have been used.)
Note that area spz is the area swept since perihelion; by the second
law, that is proportional to time since perihelion. But we defined and
so M is also proportional to time since perihelion—this is why it was
introduced.
We now have a connection between time and position in the orbit. The
catch is that Kepler's equation cannot be rearranged to isolate E; going
in the time-to-position direction requires an iteration (such as
Newton's method) or an approximate expression, such as
via the Lagrange reversion theorem. For the small ? typical of the
planets (except Pluto) such series are quite accurate with only a few
terms; one could even develop a series computing T directly from
M.[1] (http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html)
[edit]
Kepler's understanding of the laws
Kepler did not understand why his laws were correct; it was Isaac Newton
who discovered the answer to this more than fifty years later. Newton,
understanding that his third law of motion was related to Kepler's third
law of planetary motion, devised the following:
where:
P = object's sidereal period
a = object's semimajor axis
G = 6.67 × 10?11 N · m2/kg2 = the gravitational constant
m1 = mass of object 1
m2 = mass of object 2
? = mathematical constant pi
Astronomers doing celestial mechanics often use units of years, AU, G=1,
and solar masses, and with m2<<m1, this reduces to Kepler's form. SI
units may also be used directly in this
