Download Orbits

Orbits
• Observable - Mercury & Venus are always in close
proximity to the Sun, and either trail or lead the Sun
in the sky
• Observable - Mars, Jupiter, & Saturn are not
restricted to close proximity to the Sun, & are seen to
make loops in the sky during opposition with the Sun
• Observable - The Sun & planets follow a narrow
path through the sky through 12 constellations
(Zodiac). Mars, Jupiter, & Saturn move at a slower
pace through the Zodiac than the Sun, Mercury, &
Venus.
• Observable - The Moon is seen to go through
phases on an ~ 29-day cycle.
Old belief - Geocentric (EarthCentered) Model
• Ptolemaic System (after Ptolemy, 2nd century A.D.)
Geocentric
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Observable - Mercury & Venus are always in close proximity to
the Sun, and either trail or lead the Sun in the sky. Me & V orbit
the Earth, but go through epicycles. Their orbits have the same
period as the Sun’s period around the Earth
Observable - Mars, Jupiter, & Saturn are not restricted to close
proximity to the Sun, & are seen to make loops in the sky during
opposition with the Sun. Ma, J, & Sa’s have different orbital
period around the Earth than Me, V, and the Sun. These planets
also go through epicycles
Observable - The Sun & planets follow a narrow path through
the sky through 12 constellations (Zodiac). Mars, Jupiter, &
Saturn move at a slower pace through the Zodiac than the Sun,
Mercury, & Venus. The orbits of the planets and the Sun around
the Earth are coplanar. Ma, J, Sa’s orbits around the Earth are
exterior to that of the Me, V, & the Sun
Observable - The Moon is seen to go through phases on an ~
29-day cycle. The Moon orbits the earth & thus the percentage
of the Moon’s surface illuminated by the Sun with respect to an
observer on Earth changes
New Belief - Heliocentric (SunCentered) Model
• Copernicus (1473-1543): Credited for the
Heliocentric Model
• Tycho Brahe (1546-1601): Made accurate
measurements of the positions of stars & planets
• Johannes Kepler (1571-1630): interpreted Tycho’s
data. His models supported the Heliocentric model of
the solar system
• Galileo (1564-1642): made use of a telescope to
observe the phases of Venus & the moons of Jupiter
Phases of Venus
Old model. Venus
would always be in
a crescent phase
Venus is not
observed to
always be in
crescent phase
• Galileo’s observations of the moons of Jupiter was
crucial in showing that not all objects orbit the Earth.
Heliocentric
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Observable - Mercury & Venus are always in close proximity to
the Sun, and either trail or lead the Sun in the sky. M & V orbit
the Sun on orbits interior to the Earth’s orbit around the Sun.
Observable - Mars, Jupiter, & Saturn are not restricted to close
proximity to the Sun, & are seen to make loops in the sky during
opposition with the Sun. M, J, & S orbit the Sun on orbits
exterior to the Earth’s orbit around the Sun. The loops result
from Earth overtaking a planet in its orbit
Observable - The Sun & planets follow a narrow path through
the sky through 12 constellations (Zodiac). Mars, Jupiter, &
Saturn move at a slower pace through the Zodiac than the Sun,
Mercury, & Venus. The orbits of the planets around the Sun are
coplanar.
Observable - The Moon is seen to go through phases on an ~
29-day cycle. The Moon orbits the earth & thus the percentage
of the Moon’s surface illuminated by the Sun with respect to an
observer on Earth changes
The Zodiacal Constellations
Motion of the outer planets
Motion of outer planets
• Retrograde motion: reverse of the normal
direction. An apparent westward motion of a
planet with respect to the stars, caused by the
motion of the Earth
• Opposition: The position of a planet when it is
opposite the Sun in the sky
The phases of the moon
• The moon orbits
the Earth
(Period ~ 27 days)
• The observed
phases of the
Moon are due to
the relative positions
of the Moon & Sun
with respect to the
Earth
=b
=a
= 2a
Ellipse
• Ellipse: a set of all points, the
sum of whose distance from
2 fixed points (the foci) is a
constant
• Thus, by the definition,
• The ellipse is symmetrical
about the foci. Thus, by
considering the condition at
which both angles equal zero
Ellipse
• From (B), it is thus clear that
(A)
• It is also true from (B) that,
• and that
(B)
Ellipse
• Rearranging the terms in the
last two equations, squaring
them, then adding them
together,
(A)
Ellipse
• Substituting,
(A)
• We get
• And finally
• Note that, when ! = 0, we have the distance of the
pericenter,
• When ! = 0, we have the distance of the apocenter
Conic Sections
• It turns out that
• applies to any slice
through a cone.
• Ellipse ! 0 < e < 1
• Circle ! e = 0
• Parabola ! e = 1 (a = !"
• Hyperbola ! e > 1
Kepler’s Laws of Planetary Motion
• Each planet moves in an elliptical orbit about
the Sun, with the Sun at one focus of the
ellipse.
• Equal areas of a planet’s orbit around the Sun
are swept out in equal times
• The cube of the distance from the Sun, a,
divided by the square of the time, P, required
to traverse the orbit is a constant, and is the
same for every planet
Useful units:
Kepler’s 2nd Law
• Equal areas are swept out in
equal times.
• From the figure,
• The distance "r moved in
time "t is
• Element of the area swept
out in "t is
Kepler’s 2nd Law
• Dividing both sides of
• By "t,
• I.e., angular momentum =
mr2#, so Kepler’s second law
is a result of conservation of
angular momentum
Newton’s 3 laws of motion
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•
1)
Kepler’s 3 laws are empirical. Kepler did not know
the physics behind them. Newton provided the
physical explanation of Kepler’s laws
Newton’s Laws
Law of inertia or law of conservation of linear
momentum
2)
Law of force
3)
Action-reaction - balance of forces
Newton’s 3 laws of motion
3)
Action-reaction (cont). From conservation of total
linear momentum, the initial momenta of objects 1
and 2 must be equal to the final momenta
•
Law of Gravity:
Kepler’s 3rd law from Newton’s Laws
• Consider 2 masses orbiting in circular orbits about
their center of mass. The forces acting on m1 and m2
are
Kepler’s 3rd law from Newton’s Laws
• So,
• Because
• We can write
Kepler’s 3rd law from Newton’s Laws
• Substituting in for r1,
• Solving for P and adopting m2 >> m1,
Orbital Velocity
• From before, we saw that
• where $ab = area of an
ellipse. Also,
• So,
• Lastly, from before,
Orbital Velocity
• What are the components of
the velocity vector
• For an ellipse,
• such that
• For the radial velocity
component,
Orbital Velocity
• where, from differentiating
the equation for an ellipse,
we get,
• Thus,
• Rewriting the equation for an
ellipse as
• and recalling Kepler’s 3rd
law,
Orbital Velocity & Energy Equation
• We get,
• Note that the terms can be rearranged to get the
energy equation
Orbital Velocity
• Two special cases:
1) Circular motion (r = a) & M* >> m,
2) Parabolic (a = !". The escape velocity is
Orbital Properties
• Kepler’s 3rd law:
• The eccentricity of most of the orbits is ~ 0.0. I.e.,
planetary orbits are circular.
• The inclination of most of the orbits is ~ 0o.
• Bode’s Law??
Titius-Bode Rule (Bode’s Law)
• Bode’s Law:
• Plus - “Rule” developed before asteroid belt was
discovered
• Minus - No underlying physics.
Neptune (& Pluto) do not fit