Chapter 4 Gravitation and the Waltz of the Planets

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Chapter 4Gravitation and the Waltz of the Planets

• The important concepts of Chapter 4 pertain to
  orbital motion of two (or more) bodies, central
  forces, and the nature of orbits.
• 1. What we see in the sky results from the
  rotation of the Earth on its axis, the orbital
  motion of the Earth about the Sun, the orbital
  motion of the Moon about Earth, and, to a small
  extent, the gravitational effect of the Sun and
  the Moon on the Earths axis of rotation.
• 2. The motions of the Earth produce a fundamental
  frame of reference for stellar observations.

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Two great observers Hipparchus
Tycho Brahe (2nd century BC) (1546-1601 AD)
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Angle measuring devices mural quadrant cross
staff
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Nicholas Copernicus (1473-1543) revived the
heliocentric model for the solar system, where
planetary orbits are envisaged as circular for
simplicity. Even circular orbits are sufficient
for understanding the difference between
sidereal (star) period of a planet, Psid time
to orbit the Sun, and its synodic period, Psyn
time to complete a cycle of phases as viewed
from Earth. The relationship between the two is
best demonstrated by considering the amount by
which two planets A and B advance in their orbits
over the course of one day.
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Over the course of one day, planet A advances
through the angle
Planet B advances through the angle The
difference in the angles is the amount by which
planet A has gained on planet B, which is
related to its synodic period, i.e.
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In other words, Or If A is Earth, and B a
superior planet (orbits outside Earths orbit),
then For Earth P? 365.256363 days (a little
more than 365¼ days), i.e. Given two values,
the third can be found !
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The same technique can be used to relate a
planets rotation rate and orbital period to the
length of its day. The arrow is a fixed feature
on the planet.
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For example, the planet Mars has a synodic period
of 780 days, which means it returns to opposition
from the Sun every 2.14 years. But its true
orbital period is 687 days, or 1.88 years, which
means it returns to the same point in its orbit
about the Sun every 1.88 years. Some further
consequences Mercury Prot 58d.67, Psid
88d.0, Pday 176d. Venus Prot ?243d
(retrograde rotation), Psid 224d.7, Pday
?117d. Moon Prot 27d.3215, Psid 365d.2564,
Pday 29d.5306. Earth Prot 23h 56m, Psid
365d.2564, Pday 24h. Which planet has the
longest day?
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Johannes Kepler (1571-1630)
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How Kepler triangulated the orbit of Mars. He
took Tychos observations of Mars relative to
the Sun separated by the planets 687d orbital
period (with Earth at different parts of its
orbit) and used them to triangulate the location
of Mars, which was at the same point of its orbit.
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Keplers study of the orbit of Mars resulted in
his three laws of planetary motion 1. The
orbits of the planets are ellipses with the Sun
at one focus. Actually they are conic
sections. 2. The line from the Sun to a planet
sweeps out equal areas of orbit in equal time
periods. Angular momentum is conserved, i.e. mvr
constant. 3. The orbital period of a planet is
related to the semi-major axis of its orbit by P2
a3 (Harmonic Law).

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centre distance, c ae
2a
a ½ string length 2a
b
ae
c2 b2 a2
2b
?e (1 ? b2/ a2) ½
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• Isaac Newton formulated Keplers Laws into a
  model of gravitation, in which a mass attracts
  another mass with force inversely proportional to
  the square of the distance between the two, i.e.
  F 1/d2. Forces produce acceleration of an
  object proportional to its mass, i.e. F ma,
  and objects stay at rest or in constant motion in
  one direction unless acted upon by a force.

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• Objects in orbit around Earth are constantly
  falling towards the Earth. They are acted upon by
  gravity, but are in free-fall towards Earth. They
  will not fall to Earth if their transverse
  speed is large enough.

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The importance of Keplers 3rd Law is that, as
shown by Newton, the constant of proportionality
for a3 P2 contains two constants, p (pi) and G
(the gravitational constant), plus the sum of the
masses of the two co-orbiting bodies. If one can
determine orbital periods P and semi-major axes
a, then one can derive the masses of the objects
in the system either planets or stars ! For
example Jupiters mass from the Galilean
satellites.
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Astronomers try to keep the calculations simple,
so they usually omit p and G. Thus, the Newtonian
version of Keplers 3rd Law is usually written
as where the sum of the masses of the two
co-orbiting objects, M1 and M2, is calculated
in terms of the Suns mass, the orbital
semi-major axis (radius) a is calculated in
terms of the Earths distance from the Sun, the
Astronomical Unit, and the orbital period P is
expressed in Earth years. The point to be
emphasized is that a measurement of two of the
parameters permits one to calculate a value for
the third parameter. Astronomers use the
relationship to measure the masses of planets and
stars.
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Vis-Viva Equation. There is a useful relationship
for orbital speed that can be obtained from the
energy equation Solving the equation for the
velocity v gives which is the vis-viva
equation, where a is the semi-major axis of the
orbit, r is any point in the orbit, and v is the
speed in the orbit at r. Escape velocity is
attained when a ? 8, i.e. Circular orbits
apply when r a everywhere, i.e.
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Earths orbital velocity. And escape
velocity from Earth orbit is The maximum
encounter velocity for a solar system object is
the sum of the previous two values, i.e. which
is close to the encounter velocity of meteoroids
associated with Halleys Comet.
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Sending a satellite to the Sun. Here, the
situation is pictured at right, where an orbit
from Earth to the Sun will have a semi-major
axis of ½a. By Keplers 3rd Law the
orbital period is calculated as But aphelion
to perihelion constitutes exactly half an
orbit, so the time to reach the Sun is
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Astronomical Terminology Rotation. The act of
spinning on an axis. Revolution. The act of
orbiting another object. Geocentric.
Earth-centred. Heliocentric. Sun-centred. Opposi
tion. When a planet is opposite (180 from) the
Sun. Conjunction. When a planet is in the same
direction as. Typically refers to conjunction
with the Sun. Inferior planet. A planet orbiting
inside Earths orbit. Superior planet. A planet
orbiting outside Earths orbit. Prograde motion.
When a planets RA increases nightly. Retrograde
motion. When a planets RA decreases
nightly. Astronomical Unit A.U. The average
distance between Earth and the Sun. Inertia. An
objects resistance to its state of
motion. Inertial reference frame.
non-accelerated frame.
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Astronomical Terminology 2 Eccentricity. The
amount of non-circularity of an orbit, from round
(e 0.0) to very flattened (e 0.9). Semi-major
Axis. Half the length of the long axis of an
ellipse, equivalent to the radius of an
orbit. Orbital Period. The time taken for one
object to orbit another object. Synodic Period.
The time taken for an object to cycle through its
phases as viewed from Earth. Inferior planet. A
planet orbiting inside Earths orbit. Superior
planet. A planet orbiting outside Earths
orbit. Prograde motion. When a planets RA
increases nightly. Retrograde motion. When a
planets RA decreases nightly. Gravity. The force
exerted by an object on any other object in the
universe. Zero gravity. A fictional term
referring to the apparent weightlessness of an
object in free fall.
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Sample Questions

• 1. Imagine a planet moving in a perfectly
  circular orbit around the Sun. Because the orbit
  is circular, the planet is moving at a constant
  speed. Is the planet experiencing acceleration?
  Explain your answer.
• Answer Yes, it is. The planet experiences
  acceleration since it is constantly falling
  towards the Sun.

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• 2. Suppose that astronomers discovered a comet
  approaching the Sun in a hyperbolic orbit. What
  would that say about the origin of the planet?

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• Answer. Objects in hyperbolic orbits are not
  bound to the object they are orbiting.
  Astronomers would therefore conclude that the
  comet is not bound to the solar system and must
  therefore have originated from outside the
  solar system.

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• 3. Why is the term zero gravity meaningless? Is
  there a place in the universe where no
  gravitational forces exist?

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• Answer. All objects are subject to the attractive
  force of every other object, in proportion to the
  inverse square of the separation r from the other
  object. For one object to experience no outside
  gravitational forces, i.e. zero gravity, it would
  have to be an infinite distance away from every
  other object, which is not possible. So the term
  zero gravity cannot apply anywhere in the known
  universe.