Download Kepler`s Laws

Kepler’s Laws
• What are the shapes and important
properties of the planetary orbits?
• How does the speed of a planet vary as it
orbits the sun?
• How does the period of a planet's orbit
depend on its distance from the Sun?
Kepler’s First Law
• The orbits of the
planets are elliptical
(not circular) with
the Sun at one focus
of the ellipse.
• 'a' = semi-major
axis: Avg. distance
between sun and
planet
Kepler’s Second Law
Kepler determined that a planet moves
faster when near the Sun, and slower when
far from the Sun.
Planet
Faster
Sun
Slower
Kepler's Second Law
A line connecting the Sun and a planet sweeps out equal
areas in equal times.
slower
faster
Translation: planets move faster when closer to the Sun.
Kepler's Third Law
The square of a planet's orbital period is proportional to the
cube of its semi-major axis.
P2
is proportional to a3
or
P2 (in Earth years) = a3 (in A.U.)
1 A.U. = 1.5 x 108 km
Translation: The further the planet is from the
sun, the longer the period.
Correction to Kepler’s Third Law
 Earth and sun
actually rotate about
their common center of
mass
 Corresponds to a
point inside sun
 Used to detect
extrasolar planets
Why?
Kepler’s Laws provided a complete
kinematical description of planetary
motion (including the motion of planetary
satellites, like the Moon) - but why did
the planets move like that?
The Apple & the Moon
Isaac Newton realized that the motion of a
falling apple and the motion of the Moon
were both actually the same motion,
caused by the same force - the
gravitational force.
Universal Gravitation
Newton’s idea was that gravity was a
universal force acting between any two
objects.
Gravitational Field Strength
To measure the strength of the gravitational
field at any point, measure the gravitational
force, F, exerted on any “test mass”, m.
Gravitational Field Strength, g = F/m
Gravitational Field Strength
Near the surface of the Earth, g = F/m =
9.8 N/kg = 9.8 m/s2.
In general, g = GM/r2, where M is the mass
of the object creating the field, r is the
distance from the object’s center, and G =
6.67 x10-11 Nm2/kg2.
Gravitational Force
If g is the strength of the gravitational field
at some point, then the gravitational force
on an object of mass m at that point is Fgrav
= mg.
If g is the gravitational field strength at
some point (in N/kg), then the free fall
acceleration at that point is also g (in m/s2).
Gravitational Field Inside a
Planet
If you are located a distance r from the
center of a planet:
– all of the planet’s mass inside a sphere of
radius r pulls you toward the center of the
planet.
– All of the planet’s mass outside a sphere of
radius r exerts no net gravitational force on
you.
Gravitational Field Inside a
Planet
The blue-shaded part
of the planet pulls you
toward point C.
The grey-shaded part
of the planet does
not pull you at all.
Gravitational Field Inside a
Planet
Half way to the center of the planet, g has
one-half of its surface value.
At the center of the planet, g = 0 N/kg.
Kepler’s Laws are just a special case
of Newton’s Laws!
Newton explained Kepler’s Laws by solving
the law of Universal Gravitation and the law of
Motion
Ellipses are one possible solution, but there are
others (parabolas and hyperbolas)
Kepler’s Laws are just a special
case of Newton’s Laws!
Newton explained Kepler’s Laws by solving
the law of Universal Gravitation and the law of
Motion
Ellipses are one possible solution, but there are
others (parabolas and hyperbolas)
Bound and Unbound Orbits
Unbound (comet)
Unbound (galaxy-galaxy)
Bound
(planets,
binary stars)
Understanding Kepler’s Laws:
conservation of angular momentum
L = mv x r = constant
r
larger distance
smaller v
planet moves slower
smaller distance
smaller r
bigger v
planet moves faster
Understanding Kepler’s Third
Law
Newton’s generalization of Kepler’s Third Law is given by:
4p2 a3
p2 =
G(M1 + M2)
Mplanet << Msun, so 
4p2 a3
p2 =
GMsun
This has two amazing implications:
The orbital period of a planet depends only on
its distance from the sun, and this is true
whenever M1 << M2
An Astronaut and the Space Shuttle
have the same orbit!
Second Amazing Implication:
If we know the period p and the average
distance of the orbit a, we can calculate the
mass of the sun!