Transcript Motion in the Sky & Getting to know the Sky

Outline - Jan. 28, 2010
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Correct description of planetary orbits (Kepler; pgs. 72-74)
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Death of “Aristotelian” view of the heavens (Galileo; pgs. 75-78)
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Properties of astronomical objects (mass, temperature, energy)
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Newton’s laws of motion (radical departure from “Aristotelian”
physics); pgs. 121-129
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Newton’s law of gravity (134-136)
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Types of Energy (pgs. 130-134)
Correct Model of the Solar System
Johannes Kepler (1571-1630)
using Tycho Brahe’s lifetime’s worth of planetary motion data
First Law: The orbit of each planet around the
sun is an ellipse with the sun at one focus.
Second Law: As a planet moves around its
orbit, it sweeps out equal areas in equal time.
Third Law: The relationship between a planet’s
period of orbit about the sun (P) and the semimajor axis of its orbit (a) is
P2 = a3
where P is measured in (earth!) years, and a is
measured in AU
Purely empirical laws; they only tell you “what”, not “why”
Kepler’s First Law
planets move on elliptical orbits
An ellipse is a geometric figure
with two symmetry axes: major
axis (= long axis) and minor axis
(= short axis). Half the length of
the major axis is called the “semimajor axis”.
The farther apart are the two
foci, the “flatter” is the ellipse.
The closer together are the two
foci, the “rounder” is the ellipse.
A circle is a special case of an
ellipse in which the two foci are
on top of each other (in the
center of the circle).
Orbital Shapes and Distances
a = semi-major axis
b = semi-minor axis
c = distance from center of ellipse
to one focus
Shape of ellipse described by “eccentricity”:
e=c/a
Closest approach of planet to sun:
perihelion distance = a(1-e)
Farthest distance of planet from sun:
aphelion distance = a(1+e)
Kepler’s Second Law
equal areas in equal time
The areas (“square footage”)
of each of the blue triangles
are identical. The time (t) is
also identical for each triangle.
Consequence of Kepler’s
second law: planets move
faster when they are close
to the sun than when they
are far from the sun.
speed = distance / time
Kepler’s Third Law
P2 = a3
P = the time it takes a planet to orbit once around the sun,
measured in EARTH years
a = half the length of the long axis of the planet’s orbit,
measured in AU
Galileo Galilei
(1564-1642)
First person known to use a telescope to
make detailed study of the sky; a
contemporary of Kepler (8 years older)
Observations provided strong proof that
the Aristotelian (= ancient Greek)
philosophy of the heavens was wrong
Observations provided direct proof that not
all objects orbited the earth and at least
one (Venus) orbited the sun
Sun, Moon and Jupiter are not “Flawless Orbs”
Galileo observed that sun has black spotches (sunspots), moon has craters
and mountains (very “earth-like” terrain), and Jupiter has big red spot
Galilean Moons of Jupiter
(not all objects orbit the earth)
Galileo discovered the 4 largest moons of Jupiter, and
showed that they orbit Jupiter, not the earth
Venus Orbits the SUN!
Venus could not show “full” phase in Ptolemy’s model
According to Ptolemy, Venus would
never show the “full” phase
because it could never be on the
opposite side of the sky as the sun
(as seen from earth). Note: Venus
is often (mistakenly) called the
“morning star” or the “evening star”.
Phases of planets can only be seen using a
telescope. Venus shows the full range of phases
(full, gibbous, crescent, new) just like the moon.
Venus is smallest (and dimmest) in its full phase.
Phases of Venus in the “Keplerian” Model
(Venus orbits the sun, and is always closer to the sun than Earth)
Venus is in the “full” phase when it is at its farthest distance from the earth,
therefore it appears very small in the sky.
Venus is in the “new” phase when it is at its closest distance to the earth, so
the “narrow crescent” Venus looks very large (and very bright)!
Where do we go from here?
Real Physical Insight about the Universe
Want to deduce information about astronomical objects:
• Mass
• Temperature
• Luminosity and Energy
• Motion
Need a toolbox to do this:
• Fundamentals of motion and gravity (gives MASS)
• Properties of light (gives TEMPERATURE, LUMINOSITY
and MOTION towards/away from observer)
• Properties of matter (gives ENERGY production in stars)
Descriptions of Motion
(and related quantities)
Speed, s
Velocity, v
Acceleration, a
(Linear) Momentum, p = mv
Angular Momentum, l = mvr (on a curve of radius r)
“p” and “l” are examples of “conserved quantities”
When you step on the bathroom scale are you measuring your MASS?
Change in Velocity (= Acceleration) without a Change in Speed?
Any object moving with a
constant speed on a circular orbit
is being constantly accelerated
because the direction of motion
is constantly changing!
Cause of accelerations: forces!
The “why’s” of orbital motion (planets, moons) come from combining the idea of
a generic force that causes accelerations with the specific force of gravity.
Isaac Newton
1642-1727
Three Laws of Motion
N1: An object moves at a constant velocity if there is no net force
acting upon it. (“Law of Inertia”) a truly radical idea!!!!
N2: A force on an object of mass “m” produces an acceleration,
where F = ma (or, equivalently, a = F / m).
N3: For any force, there is always an equal and opposite
reaction force.
Units of force: “Newton” (N), 1 N = 1 kg x m/s2
Newton’s Explanation of Orbital Motion
Originally formulated for motion of the
moon around the earth, but works
equally well for planets orbiting the sun
Law of Gravity - All objects that have mass attract each other with a very specific
relationship for the force:
FG = (G m1 m2)/d2
where m1 and m2 are the masses of two objects (in kg), d is the distance between
them (in km), and G is “Newton’s Constant”, G = 6.67x10-11 m3/(kg x s2)
Planets orbit (instead of traveling in straight lines at
a constant speed) because a force is applied to them
(by the sun), and that force is the force of gravity.
What a minute!! What about N3??
N3: For any force, there is always an equal and opposite reaction force.
So, doesn’t that mean that the earth exerts an identical force of gravity on
the sun as the sun exerts on the earth?
Why does the earth “orbit the sun”?
Shouldn’t the sun also move in reaction to the force from the earth?
Believe it or not, this is one of the ways that astronomers have found
planets orbiting stars other than the sun!!
Newton’s Version of Kepler’s Third Law
Any two objects with masses M1 and M2 orbit about each
other according to:
Here G is Newton’s gravitational constant, P is the orbital period and a is the
distance between the two objects. If you use G = 6.67x10-11 m3/(kg x s2), then
you want P in units of seconds and a in units of meters.
If M2 is MUCH SMALLER than M1, this formula gives you a way to measure M1 (you
just need to know “P” and “a”)
See Mathematical Insight 4.3 on page 137 where the mass of the sun is calculated
using the earth’s orbit.
Warning! Book Typo! The period of the earth’s orbit is 1 year = 3.15x107 seconds.
Types of Energy
Units: Joules (J), 1 Joule = 1 kg x m2/s2
Kinetic Energy - energy of motion
Potential Energy - stored energy that can be converted into another kind of
energy at some later time (gravitational and chemical are common)
Radiative Energy - energy of light
Mass-Energy - energy contained within matter itself (E = mc2); stars power
themselves by converting mass into light!!
Thermal Energy - kinetic energy of many particles (higher the temperature,
the faster the speed, and the greater the kinetic energy)
Temperature
Units: Kelvin (K)
Celsius scale is defined such
that water freezes at 0oC and
water boils at 100oC
Kelvin scale is defined by how
cold it needs to be for
molecules to be unable to
move/vibrate (“Absolute Zero”)
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