Tycho was a Danish nobleman interested in astronomy. In 1572 a "new star"
(in today's language, a nova) appeared in the sky, not far from Polaris,
outshining all others. Tycho carefully measured its position, then
measured it again 12 hours later, when the the rotation of the Earth had
moved the observing point to the other side of the Earth. Such a move was
already known to shift the position of the Moon in the sky, helping
astronomers estimate its distance. The position of the "new star" did not
change, suggesting it was much more distant than the Moon
This event so impressed young Tycho that he resolved to devote himself to
astronomy. The king of Denmark supported him and gave him the island of
Hven to build an observatory, with the taxes of the island providing him
with the funding. The telescope had not yet been invented, and all
measurements were done by eye, aided by sights (similar to those used on
guns) which could be slid around circles, marked in degrees. Tycho
extended such methods to their ultimate limit, the resolution of the human
eye, and his star charts were far more accurate than any earlier ones. He
even measured and took into account the very slight shift of star
positions near the horizon, due to the bending of light in the Earth's
atmosphere, similar to its bending in glass or water. And his observations
of the planets became the most stringent test of the theories of
Copernicus and Ptolemy. Concerning those theories,
Tycho believed that all planets revolved around the Sun, but the Sun
circled Earth. That view might have suited Denmark's Protestant church,
for Martin Luther, founder of the Protestant doctrine, had rejected the
views of Copernicus (who lived at the same time). Tycho's manners,
however, were arrogant, and the residents of Hven complained about him, so
that after the death of the king who was Tycho's patron, Tycho was forced
to leave Denmark.
He settled in 1599 in Prague--now the Czech capital, then the site
of the court of the German emperor Rudolf--and there he became court
astronomer. It was in Prague, too, where a German astronomer named
Johannes Kepler was hired by Tycho to carry out his calculations. When in
1601 Tycho suddenly died, it was Kepler who continued his work.
Johannes Kepler (1571-1630)
Kepler had studied astronomy long before he met Tycho: he favored the
Copernican world-view and corresponded with Galileo.
Tycho's observations included some very accurate measurements of the
position of the planet Mars, which did not agree with either Ptolemy or
Copernicus. When Tycho died, Kepler got hold of those observations and
tried to puzzle them out. In 1609, the same magic year when Galileo first
turned his telescope towards the heavens, Kepler caught a glimpse of what
he thought might be the answer. That was when he published his first two
laws of planetary motion:
Planets move along ellipses, with the Sun at one focus.
The line from the Sun to the planet covers equal areas in equal times.
Each of these statement requires some explanation.
Ellipses
The ellipse, the shape of a flattened circle, was well known to the
ancient Greeks. It belonged to the family of "conic sections," of curves
produced by the intersections of a plane and a cone.
As the drawing on the left shows, when that plane is...
perpendicular to the axis of the cone, the result is a circle.
moderately inclined, an ellipse.
inclined so much that it is parallel to one side of the cone, a
parabola.
inclined even more, a hyperbola.
Kepler's First Law
The Ellipse in Polar Coordinates
If all the values of (r,f) of a curve are related by some equation which
can be symbolically written r = r(f) then the function r(f) is said to be the equation of the line, in
polar coordinates. The simplest function is a constant number a, giving
the line r = a
The value of r equals a for any value of f. That gives a circle around the
origin, its radius equal to a, shown in the drawing on the right above.
The Ellipse
Consider next the curve whose equation is
r = a(1- e2)/(1+ e cos f)
where the eccentricity e is a number between 0 and 1. If e = 0, this
is clearly the circle encountered earlier.
How about other values? The function cos f represents a
wave-like behavior (picture below), and as f goes through a full circle,
it goes down, from +1 to 0, then -1, then up again to 0 and +1. The
denominator also rises and falls as a wave, and it is smallest when cos f
= -1 . Here is the table of the main values (360 is in parentheses,
because it represents the same direction as 0 degrees):
f degrees
90
180
270
(360)
cos f
1
0
-1
0
1
1 + cos f
1 + e
1
1 - e
1
1 + e
As long as e is less than 1, the denominator is always positive. It is
never zero, so that for any f one can name, one can always find a suitable
r. In other words, the curve goes completely around the origin, it is
closed.
The expression (1 - e2) can be factored--that is, written as
two expressions multiplied by each other ("the product of two
expressions").
1 - e2 = (1 - e)(1 + e)
At some of the points on the above table, either (1 - e) or (1 + e)
cancels the denominator, giving:
f degrees
0
90
180
270
(360)
r
a(1 - e)
a(1 - e2)
a(1 + e)
a(1 - e2)
a(1 - e)
As mentioned in the preceding section a second focus O' can be drawn
symmetrical to O, and the ellipse can be defined (its original definition,
in fact) as the collection of points for which the sum R1+R2 of their
distances from O and O' is always the same The longest
dimension of the ellipse, its width AB along the line connecting the two
foci, is its "major axis." Suppose (R1,R2) are the distances of A from the
foci O and O'. Then R1 = OA =a(1 - e) is the smallest distance of the
ellipse from O, R2 = O'A = OB (by symmetry) is the largest and therefore
equals a(1 + e). But, OA + OB = AB, hence
AB = a(1 - e) + a(1 + e) = 2a
Hence the quantity a in the equation of the ellipse is known as
semi-major axis. We can now state Kepler's law more precisely as "the
square of the orbital period T is proportional to the cube of the
semi-major axis a of its orbit around the Sun." The two
quantities (a,e) completely define the ellipse. When that ellipse is the
orbit of a planet or a satellite, they form two of the six orbital
elements which define the state of the orbiting body. A third element, the
mean anomaly M, specifies the position of the planet or satellite along
the orbit, and the remaining three define the orientation of the orbit in
3-dimensional space.
Refining the First Law
Kepler's first law states:
"The orbit of a planet is an ellipse, with the Sun at one focus"
Actually, this is not 100% accurate. Imagine the planet magically getting
heavier and heavier, while the Sun is getting lighter and lighter. At some
point both of them would be equally heavy: could we then say which is
orbiting around which?
To be fully accurate, the first law should have placed the
focus of the orbital ellipse at the center of gravity of the planet-Sun
system. (The center of gravity will be defined later, but intuitively, if
the masses are very unequal, as with a planet and the Sun, it lies close
to the center of the heavier object.) Because the Sun is so much heavier
than the Mars, the effect on the orbit of Mars (which Kepler studied) was
too small to be noted by him. Nevertheless, the Sun also moves in response
to motions of its planets, and motions of this type have become an
important tool in the search for planets outside the solar system.
An Earth-sized planet orbiting a distant star would be far
too dim to be seen with any earthly telescope, especially against the
glare of its sun, the star itself. However, as the planet goes around the
orbit, its star also moves in a mirror image orbit around the common
center of gravity. It is a much smaller orbit and a much slower motion,
because the center of gravity is very close to the center of that star (in
the Earth-Sun system, it is inside the Sun), but it can still be detected
by subtle variations of the starlight.
Recently a few such planets have been found, but most
are Jupiter-sized and none seems suitable for life.
Bodies in the solar system can also move in other conic sections, in
parabolas or hyperbolas, whose equations resemble that of an ellipse, but
have e equal to 1 or larger. Properly speaking, these orbits do not have a
"semi-major axis," because they are not limited in size like ellipses but
extend to infinity. Instead, we must write
r = p/(1 + e cos f)
which doe not raise any problems at e=1. Such bodies are not bound to
the Sun, but are free to escape it. The denominator in the equation of the
trajectory then becomes zero for some values of f, making its r infinite,
and as the moving body approaches those values, it moves further and
further away, with no limit. Comets in general have an eccentricity e
close to 1, suggesting they have come from the very distant fringes of the
solar system. The space probe Voyager 2 has e > 1 and is on its way out
of the solar system, never to return.
Kepler's Second Law
The ellipse traced by a planet around the Sun has a symmetric shape, but
the motion is not symmetric.
Think of a stone thrown upwards: as it rises it loses speed, then for an
instant, at the top of the trajectory, it moves very slowly, and finally
it comes down, gathering speed again. The motion of a planet around the
Sun or of a scientific satellite around Earth follows different equations
(though a connection exists), but in many respect it resembles that of the
stone.
That is most evident if the orbit is elongated, that is, its eccentricity
is not far from 1. As the planet or satellite rises in its orbit, it slows
down, then as it returns, it speeds up again, moving at its fastest during
its closest approach. That point of the orbit is called perihelion for a
planet ("helios" is the Sun) and perigee for an Earth satellite ("gee"
from "geo", denoting Earth-related).
After studying actual observations, mainly of Mars, Kepler proposed the
following prescription for predicting the speeding-up and the
slowing-down. Let a line ("radius vector") be drawn from the center of the
Sun to the planet (or from the center of Earth to the satellite).
Kepler's Second law states:
"The radius vector sweeps equal areas in equal times"
As an example, let the drawing on
the right represent the orbit of an Earth satellite, and let AB and CD be
the portions of the orbit covered in 3 hours near apogee and near perigee,
respectively. If then O is the Earth's center, the shaded areas OAB and
OCD are equal. What it means, obviously, is that CD is much longer than
AB, because near perigee the satellite moves much faster and it covers a
much greater distance in 3 hours.
Energy
Energy may be loosely defined as anything that can make a machine move.
The forms of energy which power our machines are usually electricity or
heat; light is another form, converted into electricity by the solar cells
which power most satellites.
Gravity can also provide energy. The wheels of grandfather clocks are
turned by weights which gradually descend to the bottom of the clock, at
which point they have to be cranked up again, or else the clock stops.
Thomas Jefferson, at his home near Charlottesville, Virginia, had a clock
whose weights (hanging on the side of the room) were cannonballs strung on
a rope, and to give the clock a 7-day range, a hole was cut in the floor
allowing the balls to descend to the basement.
When a weight or cannonball is raised against the force of gravity, it has
potential energy--energy by virtue of its position, proportional to the
height to which it was raised. If the weight is dropped, it loses height
and potential energy, but gains speed and kinetic energy, the energy due
to speed of motion. Kinetic energy can be converted back to potential, as
happens to a roller coaster after it passes the bottom of a dip and climbs
up again.
A similar change occurs when a stone is thrown upwards with some velocity
v. If its mass is m (mass will be defined later, for now view it as
something related to weight), its kinetic energy can be shown to be
1/2 mv2
As it rises, v and the kinetic energy decrease, but this is matched by the
growth of the potential energy
h m g
where h is the height in meters and g is a constant measuring the strength
of the force of gravity: if m is in kilograms, h in meters and v in
meters-per-second (written m/sec; walking speed is about 1-2m/sec), g is
about 9.81.
The sum of the two is the total energy E and stays constant:
E = 1/2 mv2 + h m g = constant
As the stone rises, the kinetic part of its energy gets smaller and
smaller, becoming zero when it reaches its highest point, where for a
brief instant v = 0. On the downward trip, the opposite changes take
place. In a later section we will come back to that formula and to the
concept of energy.
For a satellite of mass m orbiting Earth (or for a planet around the Sun)
a similar formula exists:
E = 1/2 mv2 - k m/r = constant
Here k is some other constant--actually, related to g, because both
constants reflect the strength of the Earth's gravity (the exact value is
k = gR2, where R is the radius of the Earth, in meters). Don't let the
minus sign confuse you: as the satellite rises, r increases, k m/r becomes
smaller, but -k m/r becomes bigger, it is less negative than near Earth.
This equation shows why the satellite's speed decreases as it moves away
and grows as it comes back.
Suppose the satellite has just enough velocity to escape Earth's gravity
altogether (the "escape velocity" V). Then far from Earth, where k m/r is
close to zero, its kinetic energy would also be exhausted, that is, v = 0.
Since the sum E is the same everywhere, this suggests that for the space
probe which just barely escapes the Earth's gravity, E=0. From that
V2 = 2k/R = 2 g R
With g = 9.81 and R =6 371 000 meter on finds V to be about 11200 m/sec.
The Mean Anomaly
Earlier it was stated that a third orbital element is needed to specify
where in its orbit is the satellite located. Since the equation of the
orbital ellipse is
r = a(1 - e2)/(1 + e cos f)
each value of the angle f --called the "true anomaly"--specifies a
position along the orbit. One could therefore use the true anomaly as
third orbital element.
The true anomaly f varies periodically around the orbit, quickly near
perigee and slowly near apogee. Kepler's second law tells us everything
about this variation and should allow us to obtain a formula that gives
the way f varies with time t. Unfortunately, no neat way exists for
expressing that formula.
The simplest way of expressing f is to use two auxiliary angles, which
like f increase by 360 degrees each orbit, the "eccentric anomaly" E (the
letter here has nothing to do with energy) and the "mean anomaly" M; an
equation then exists connecting f and E, and another that connects E and
M. The great virtue of M is that it grows in proportion to the time t:
M = M(0) + nt
where M(0) is the value of M when t = 0 and n is a constant (related to
the constant appearing in Kepler's third law). The mean anomaly is what is
counted as the third orbital element.
If one wishes to predict the position of a satellite in its orbit at some
time t, assuming the elliptical motion of Kepler's laws is good enough for
that prediction (neglecting the pull of the moon, friction of the upper
atmosphere etc. ) the first step is to derive M from the above formula.
Then E is derived from E, and finally f from E, tasks which electronic
computers handle quite easily (though at one time, those calculations were
done on paper, not nearly as quickly or easily). The formula for r then
gives the position of the satellite in its orbit; all that the computation
requires are the elements a, e and M(0), the mean anomaly at t=0.
Below, a drawing of the orbit of Mars, from Kepler's writings
Kepler's Third Law
Kepler's Third Law States:
The square of the orbital period of a planet is directly proportional
to the cube of the semi-major axis of its orbit.
This captures the relationship between the distance of planets from the
Sun, and their orbital periods.
All these intersections are easily produced by a flashlight in a
moderately dark room . The flashlight creates a cone of light and when
that cone hits a wall, the shape produced is a conic section--the
intersection of the cone of light with the flat wall. The axis of the
flashlight is also the axis of the cone of light. Aim the beam
perpendicular to the wall to get a circle of light. Slant the beam: an
ellipse. Slant further, to where the closing point of the ellipse is very,
very far: a parabola. Slant even more, to where the two edges of the patch
of light not only fail to meet again, but seem to head in completely
different directions: a hyperbola.
After Tycho's death, Kepler became the court astronomer, although the
superstitious emperor was more interested in astrology than in the
structure of the solar system. In 1619 Kepler published his third law: the
square of the orbital period T is proportional to the cube of the mean
distance a from the Sun (half the sum of greatest and smallest distances).
In formula form
T2= k a3
with k some constant number, the same for all planets. Suppose we measure
all distances in "astronomical units" or AUs, with 1 AU the mean distance
between the Earth and the Sun.
Then if a = 1 AU, T is one year, and k with these units just equals 1,
i.e. T2= a3.
Applying now the formula to any other planet, if T is known from the
observations of many years, the planet's a, its mean distance from the
Sun, is readily derived. Finding the value of 1 AU in miles or kilometers,
that is, finding the actual scale of the solar system, is not easy. Our
best values nowadays are the ones provided by space-age tools, by
radar-ranging of Venus and by planetary space probes; to a good
approximation, 1 AU = 150 000 000 km.
Kepler's 3rd Law
T in years, a in astronomical units; then T2 = a3
Discrepancies are from limited accuracy
Planet
Period T
Dist. a fr. Sun
T2
a3
Mercury
0.241
0.387
0.05808
0.05796
Venus
0.616
0.723
0.37946
0.37793
Earth
1
1
1
1
Mars
1.88
1.524
3.5344
3.5396
Jupiter
11.9
5.203
141.61
140.85
Saturn
29.5
9.539
870.25
867.98
Uranus
84.0
19.191
7056
7068
Neptune
165.0
30.071
27225
27192
Pluto
248.0
39.457
61504
61429
Not only were Kepler's laws confirmed and explained by later scientists,
but they apply to any orbital system of two bodies--even artificial
satellites in orbit around the Earth. The constant k' for artificial
satellites differs from k obtained for planets (but is the same for any
satellite). By Kepler's formula
T = SQRT (k' a3)
where SQRT stands for "square root of" (the world-wide web does not offer
more specific symbols). If T is measured in seconds and a in Earth radii
(1 RE = 6371 km = 3960 miles)
T = 5063 SQRT (a3)
Besides the Moon, Earth now has many artificial satellites, put up by us
earthlings for a variety of purposes. The calculation applied by Newton to
the Moon can also be used for them.
Orbital velocity
Suppose the Earth were a perfect sphere of radius 1 RE = 6 317 000 meters
and had no atmosphere. In principle, a satellite could then orbit just
above its surface. The Earth would pull it downwards with a force F = mg,
and because of the direction of this force, any accelerations would be in
the up-down direction, too.
If the satellite is in a stable circular orbit and its velocity is V, then
F supplies just the right amount of pull to keep the motion going. Which
means
mg = F = mV2/RE
Dividing both sides by m shows that the mass of the satellite does not
matter, and leaves
V2/RE = g Multiplication of both sides by RE: gives
This is the velocity required by the satellite to stay in its orbit ("1"
in the drawing). Any slower and it loses altitude and hits the Earth
("2"), any faster and it rises to greater distance ("3"). For comparison,
a jetliner flies at about 250 m/sec, a rifle bullet at about 600 m/sec. We
again need a notation for square root. Since the HTML language does not
provide one, we use the notation SQRT found in some computer languages.
The square root of 2, for instance, can be written
SQRT(2) = 1. 41412. . .
If the speed V of our satellite is only moderately greater than Vo curve
"3" will be part of a Keplerian ellipse and will ultimately turn back
towards Earth. If however V is greater than 1. 4142. . .times Vo the
satellite has attained escape velocity and will never come back: this
comes to about 11.2 km/sec.
Kepler's Third Law for
Earth Satellites
The velocity for a circular Earth orbit at any other distance r is
similarly calculated, but one must take into account that the force of
gravity is weaker at greater distances, by a factor (RE/r)2.
We then get
V2/r = g (RE r)2 = g RE2/r2
Let T be the orbital period, in seconds. Then (as noted earlier), the
distance 2 pr covered in one orbit equals VT
VT
= 2 p r
V
= 2 p r/T
V2
= 4 p2r2/T2
>
V2/r
= 4 p2r/T2
and by the earlier equality
4 p2r/T2 = g RE2/r2
Get rid of fractions by multiplying both sides by r2T2
4 p2r3 = g RE2 T2
To better see what we have, divide both sides by g RE2, isolating T2:
T2 = (4p2/g RE2) r3
What's inside the brackets is just a number. The rest tells a simple
message--T2 is proportional to r3, the orbital period squared is
proportional to the distance cubes. This is Kepler's 3rd law, for the
special case of circular orbits around Earth.
Applying Kepler's Third Law
r circular orbits around Earth, we found
T2 = (4p2/g RE2) r3
with T in seconds and r in meters. The distance of a satellite from the
center of Earth in meters is an inconveniently big number, even before we
raise it to the 3rd power. We may however multiply the expression on the
right by (RE3/RE3) = 1 and then rearrange the terms:
T2 = (4p2/g RE2) (RE3/RE3) r3 = (4p2RE/g) (r/RE)3
The ratio r' = (r/RE) is the orbital distance measured in units of the
Earth's radius. That number is usually between 1 (at the Earth's surface,
r = RE) and 60 (at the Moon's orbit, r ~ 60 RE).
Also, that ratio is always the same, whether r and RE are in meters, yards
or nautical miles, as long as both r and RE are measured in the same
units.
The other term is calculated below, with multiplication denoted by blank
spaces between parentheses; you may check it with your calculator.
This is the practical form of Kepler's 3rd law for Earth satellites. Our
imagined satellite skimming the surface of the Earth (r' = 1) has a period
T = 5063.5 sec = (5063.5/60) minutes = 84.4 minutes
The space shuttle must clear the atmosphere and goes a bit higher.
Say it orbits at
r' = 1. 05, with SQRT(r') = 1. 0247
. Then T = (5063.5) (1.05) (1.0247) = 5448 seconds = 90.8 minutes
International communication satellites are in the equatorial plane of the
Earth and have orbits with a 24-hour period. As the Earth rotates, they
keep pace with it and always stay above the same spot. What is their
distance?
Here T is known and we need to find r':
If all numbers on the last line are equal, their squares are equal, too
(r')3 = (17.0632)2 = 291.156
Now you need a calculator able to derive cubic roots (or else, the 1/3 =
0.333. . . power).
This gives r' = 6.628 earth radii
as the distance of "synchronous" satellites.
The satellites of the global positioning system (GPS), by which a small,
handheld instument can tell one's location on the globe with amazing
accuracy, are in 12-hour orbits.