The Solar System
 How can we determine the size of the Solar system,
and the sizes of planets, moons etc?
 Some simple geometric arguments allowed even the
ancient astronomers to get good estimates
Basic Geometry: the skinny triangle
 Objects in the sky have a pretty small angular size:
that is, they aren’t very big/wide
 The moon is only 0.5 degree diameter
 Most celestial objects have sizes of about 1 “ (1
arcsecond) or less
 1 arcmin = 1/60 degree
 1 arcsec = 1/60 arcmin = 1/3600 degree
 For angles  where <<1 radian we can
approximate sin~
 E.g. for =1 deg=0.017453 rad,
sin=0.017452
 Thus we can relate an object’s physical size D, to
its distance r and angular size :
D


2r 360
Size and Shape of Earth
 Evidence that the Earth is round:
 At sea, land at sea level disappears before hills
 Ship hulls vanish before their masts
 The altitude of stars depends on latitude
 The shadow of a lunar eclipse is always circular
 Eratosthenes calculated the size of the Earth using a
simple geometric argument
 Compare the height of the Sun at two different
locations, at the same time.
 Result was very close to the known value today
 Distance to the Moon
 Eclipses show the Moon is between Earth/ Sun
 Lunar eclipses can be used to determine the
distance to the moon
 Since we know the diameter of the Earth
(13000 km), and the angular size of the Sun
(0.53 deg) we can work out the size of
Earth’s shadow.
 Comparison with the angular size of the
shadow on the moon gives the distance to
the moon, about 350,000 km.
 Distance to the Sun:
 Aristarchos used another simple geometric
argument, based on the phases of the Moon
 When moon is in quarter phase, Earth-MoonSun must form a right-triangle
 By measuring the angle between the Moon
and Sun, as viewed from Earth, you can solve
the triangle
 The angle is 89.75 degrees, very difficult to
measure accurately. The distance to the
Sun is then 350,000/cos(89.75). Since the
angle is close to 90 deg, the cosine is close
to zero. A small error in measurement leads
to a very wrong answer.
Planetary Motions
 Planets are observed to move relative to the
background stars
 Motion is generally regular, but sometimes shows
retrograde motion that was very difficult to
explain in geocentric theories
 Led to use of epicycles
 In a heliocentric theory retrograde motion is a
natural consequence of the inner planets orbiting
more quickly than the outer planets
 Distances to planets can be determined using simple
geometric calculations
 Interior: measure angle between planet and the
Sun at greatest elongation (when they are
farthest apart)
 Exterior: measure same angle, separated in time
by planet’s orbital period.
 Copernicus had excellent values for all of the
known planets, but they were less precise for
the outermost ones (Jupiter and Saturn)
because their periods are long enough that
fewer oppositions could be observed
Kepler’s Laws
 Based on very accurate orbital data obtained by Tycho
Brahe, Kepler derived three laws of orbital motion:
1. A planet orbits the Sun in an ellipse, with
the Sun at one focus
2. A line connecting a planet to the Sun sweeps
out equal areas in equal time intervals
3. P2=a3, where P is the period and a is the
average distance from the Sun.
 Some useful equations (valid when M>>m):
4 2 a 3
G( M  m)
General form of K3:
P2 
Energy equation:
E
Vis-Viva equation:
1 1 
v 2 (r )  2GM   
 r 2a 
GMm m 2 GMm
 v 
2a
2
r
In all of these:
M is the mass of the heavier body
m is the mass of the smaller body
a is the size of the semimajor axis
v is the orbital velocity
r is the distance between M and m
P is the orbital period
E is the total energy