47
Planets:
nine planets in solar system, including Earth
( the Moon is not a planet)
five planets can be seen with naked eye
-
-
- in order of increasing distance from Sun: Mercury , Venus, Mars, Jupiter, Saturn
Heliocentric Model of Solar System
-
“Helios” – Greek for Sun
-

model of Solar system with Sun at center
-
 Earth and other planets revolve on orbits around Sun
due to Nicolaus Copernicus (Polish, 15 th
century)
-
-
-
Geocentric model: opposing model in which all celestial objects revolve around
Earth accepted until 16 th
century originally provided simplest explanation for diurnal motion
 eg. the Celestial Sphere is a geocentric model of the Universe -
Planets are celestial objects
-

planets undergo diurnal motion as Celestial Sphere ( CS ) rotates westward
AND: Planets revolve around Sun
-

planets also move on CS with respect to stars, ie.
like Sun and Moon, planets are
not fixed on CS with respect to stars
-
in Copernicus’ heliocentric model planetary orbits are circular , with Sun at center
- orbital size, r: radius of circular orbit = distance of planet from Sun
-
orbital velocity, v (orbital speed ) : planets on smaller orbits move along orbit with larger velocity, v , than planets on larger orbits
- v constant for a particular planet
-
orbital period (P): time for planet to complete one revolution of Sun
- apply d = vt formula for object of constant velocity: t=d/v
where: d = distance traveled = circumference of circular orbit = 2πr
t = orbital period, P

48
-

P = (orbital circumference ) / v = 2
 r/v
-

planets closer to Sun have smaller r AND larger v

have much smaller P
-
-
eg. Saturn: r = 10 Astronomical Units ( AU), P = 30 years

orbital velocity, v orb
, of Saturn ≈ 1/3 that of Earth
-

planets closer to Sun move more rapidly across CS than planets further
from Sun
-
 inner planets always overtaking outer planets
-
all orbits lie in almost same plane
-

planetary paths on CS lie close to plane of ecliptic (plane of Earth’s orbit)
-

planets are zodiacal objects
-
all planets revolve Eastward , like Earth
- ie.
counter-clockwise ( CCW ) as viewed from above Earth’s North pole looking
down onto plane of orbits
-
 generally , planets move Eastward on CS with respect to background stars
- Prograde : normal direction of planets’ motion on CS (Eastward)
Retrograde motion:
occassionally a planet appears to move westward with respect to background stars on
-
-
CS for a while

ie. in opposite direction of normal prograde motion
occurs only when Earth is overtaking planet, or planet is overtaking Earth
-
 due to changing direction of line of sight from Earth to planet to background
stars
-
-
 retrograde motion is an illusion due to planet’s position on CS being judged with
respect to background stars
-

planets do not actually reverse orbital direction!
Historical significance:
-
- Heliocentric model provides a much more simple explanation for retrograde motion that the Geocentric model can
-

original reason for abandoning Geocentric model for Heliocentric model in 16 th century
Range of planetary motion:

Elongation (

): angular distance from Sun to planet on CS ( ie.
as viewed from
-
Earth)
- measured East or West, whichever gives smallest angle
-
-
 changes with time as planet moves along orbit
- Greatest elongation: maximum absolute value of

-

Observational significance: planet most easily observable near greatest

- Eastern elongation: elongation measured from Sun eastward
-

when eastern

< 90 o planet is in western sky in evening
- Western elongation: elongation measured from Sun westward
-
 when western

< 90 o planet is in eastern sky in morning
-
-
Conjunction: a planet is in conjunction when

~ 0 o
- planet transits observer’s meridian near noon
-

planet not observable
Opposition: a planet is in opposition when

= 180 o
- largest value of ε a planet can have
- planet transits observer’s meridian near midnight
-

planet most easily observable
-
-
-
-
-
Inferior planets: planets closer to Sun than Earth
- two among naked-eye planets: Mercury, Venus
- have smaller orbital radius ( r ) than Earth
 greatest elongation < 90 o

planet can never be in opposition

is never near transit around midnight
 never very easily observable

have two conjunctions each orbit
- Inferior conjunction:

~ 0 o when planet in front of Sun
- Superior conjunction:

~ 0 o when planet behind Sun
-
-
-
Planetary phase: portion of planet illuminated by Sun, as seen from Earth
depends on
 
phase changes with time
same as lunar phase
- for inferior planet : phase is new at inferior conjunction
phase is full at superior conjunction
phase is quarter at greatest ε
49

-
-
-
Superior planets: planets farther from Sun than Earth
- three among naked-eye planets: Mars, Jupiter, Saturn
- have larger orbital radius ( r ) than Earth
 greatest elongation = 180 o

planet in opposition once each orbit

is near transit around midnight

most observable

have one conjunction each orbit
-

~ 0 o when planet behind Sun
Orbital periods of planets
Orbital period ( P orbital
): true orbital period
= time interval for planet to complete one full orbit (360 o
)
- linear orbital circumference = 2
 r (m), where r is radius of orbit
-
- linear orbital velocity,
-

P orbital
= 2
 r / v (s) v (m s -1 )
-
- Earth also revolving around Sun

P orbital difficult to measure
Synodic period ( P synodic
) : time interval for planet to return to same elongation ,

, on
Celestial Sphere ( CS )
-
-
eg.
time interval between successive oppositions of a superior planet
For inferior planets: P synodic
> P orbital
For superior planets: P synodic
< P orbital
Copernicus’
Heliocentric model of solar system:
-
-
planetary orbits are perfect circles with Sun at center planets’ orbital velocity is constant

Heliocentrism an important step forward
BUT Copernicus’ model did not correctly predict positions of planets
-

Copernicus’ model inaccurate
50

Johannes Kepler
-
(German, 16 th
century): modified Copernicus’ model

improved Heliocentric model
-
Kepler’s modifications in form of three laws of planetary motion
Kepler’s Law I:
The shapes of orbits
Planetary orbits are elliptical with Sun at one focus of ellipse
ellipse: “oval” type shape characterized by a major axis along the maximum dimension and a minor axis along the minimum dimension
- center is where major and minor axes cross
- - semi-major axis, a : maximum distance from center to ellipse
- - semi-minor axis, b : minimum distance from center to ellipse
-
-
- foci: two special points inside of ellipse, lying on major axis , equidistant
from center such that the sum of the distances from any point on the ellipse to
the foci is constant
-
-
- Sun located at one focus ( not at the center ), other focus empty
- Note: length of semi-major axis ,
points on ellipse to either focus a , also equals average distance from all
-
-
-
-
-
-
-
- eccentricity, e: measure of elongation or ellipticity
- e
2
= (1b
2
/ a
2
)
- planetary orbits have very small e ( e <0.1)

orbits almost circular , only
slightly eccentric
-
-
- Note: a circle is a special case of an ellipse where:
a=b
both foci are at the center -
- e =0
-
-
Orbital eccentricity, e > 0

distance from Sun to planet changes throughout orbit
-
 size of orbit specified by semi-major axis, a = average distance from Sun to
-
-
planet
 perihelion: point on orbit where planet closest to Sun
-
 aphelion: point on orbit where planet furthest from Sun
-
 perihelion and aphelion lie at points where semi-major axis intersects locus
51

Kepler’s Law II (Law of equal areas): orbital velocity
as planet revolves, an imaginary line from center of planet to center of Sun sweeps out equal areas in equal time intervals (see diagram below)
(due to conservation of angular momentum ) -
-
-

orbital velocity, v , not constant throughout orbit:
-
 v largest at perihelio n and smallest at aphelion
makes sense : planet falls toward Sun and therefore picks up speed as it approaches perihelion
Kepler’s Law III:
Relation between orbital size and orbital period
-
let P = orbital period expressed in orbital Earth years let a = semi-major axis of orbit = average distance of planet from Sun, expressed in
-
Astronomical Units (AU)
-
-
 P 2 =a 3
-CAUTION: formula only valid if P in Earth years and a in AU and planet
orbiting the Sun (not another star)
-
 P increases disproportionately as a increases
eg. Saturn: a = 10 AU

P = 10
3/2 
30 years
52

IMPORTANT NOTES ON KEPLER’S LAWS:
1) Kepler’s laws are general : they apply to any body freely orbiting another
eg.
Moon and artificial satellites orbiting Earth obey Kepler’s laws
2)
Kepler’s laws are empirical : they describe how orbiting bodies are observed to
behave, but not why they behave as they do
-
-
Sir Isaac Newton (17 th Century, England)
Significance: Laws of gravity and motion explain why bodies orbit one another and why they obey
Kepler’s Laws
Definitions:
-
Mass (M, m): amount of matter in an object
metric units: grams (g), kilograms (kg)
Quantities related to Mass:
-
weight : amount of force on a mass due to gravity
corresponds to mass , but is not the same as mass
-

mass is an intrinsic property of an object, weight depends on force of
gravity
metric units: Newtons (N)
-
volume (V): amount of space an object occupies
-
metric units: cubic cm (cm
3
), cubic m (m
3
for spherical object of radius R: V=4

R
3
/3
)
-
density (

): mass / volume ( M/V )
metric units: kg/m
3
(or kg m
-3
)
-
-
- note consistency of units: kg combine with m

two objects of same mass have different volumes if density differs
-
Significance: planets have Mass
Speed: Rate of change in position
-
metric units: m/s scalar quantity  expressed by a number only
53

-

only expresses magnitude of rate of change in position
Velocity (v): Rate of change AND direction of change in position
 ie.
speed in a given direction
vector quantity

expressed by a number (magnitude) and a direction
-
metric units: m/s (or m s if v=0

object at rest
-1 ) + direction
Acceleration (a): Magnitude of rate of change AND direction of change in velocity ( v )
-
-
-
vector quantity: has magnitude and direction metric units: (m/s)/s (or m/s
2
, m s
-2
) + direction sign of a : a>0

object’s velocity increasing
a <0

object’s velocity decreasing

“deceleration”
a may be due to change in speed only when direction is constant
-
 linear acceleration: a is parallel to v
a may be due to change in direction only when speed is constant
-
 centripetal acceleration: a c perpendicular to v
-
- - note direction of vectors:
- - v directed tangent to instantaneous point on trajectory
- - a c directed toward center of circular orbit
-

an object whose direction of travel is changing is accelerating in a direction perpendicular to the tangent to its path, even if its speed is constant
if a=0
 v is constant
 speed is constant AND direction of travel is constant
-
 ie. object is traveling in a straight line at constant speed
-
-
 planets travel on curved paths
 direction always changing
 always accelerating toward center of orbit (even if speed is constant)
-
 ie.
orbiting objects always have non-zero a
-
54

55
Force (F): a push or pull on an object that causes it to accelerate
-
-
vector quantity: has magnitude and direction metric units: Newtons (N) + direction eg. weight is the force on an object due to gravity
Newton’s Laws of Motion
Newton’s Law I:
Law of Inertia
Inertia: resistance to being accelerated by an external force, F
amount of inertia = mass (M) of object
-

mass measures object’s resistance to changing speed or direction under a push
or pull
-
an object with inertia ( ie.
M >0) does not accelerate ( ie.
has a =0) unless acted on by
a net external force ( F )
-

an object remains at rest , or remains at constant velocity (v) unless acted on by
a net external force
-
-
 planets have non-zero a

must be experiencing a net non-zero F
Newton’s Law II: Equation of motion
how an object responds to a net external force , F
-
a = F/m

ie. force causes acceleration where: a = acceleration
F = force
m = mass
-
Units in Astronomy: m in kg , F in N  a in m/s 2
Direction of a: a and F both vector quantities
 a is in same direction as (parallel to) F
 a || F
if F=0
 a=0
-
-
- ie. objects do not accelerate unless pushed
-

same as
Newton’s Law I
Common form of Law: F=ma
Weight: if F is due to gravity

F referred to as
“weight”
-
-
Planets: have F toward center of orbit due to
Sun’s gravity

56
-

have centripetal acceleration a c
=F c
/m where m is planet’s mass
-

F c
= centripetal force , directed toward center of orbit
-

F c
= ma c
-

force needed to keep object on circular orbit rather than flying off on a
tangent due to inertia
- F c
is due to Sun at center of orbit
Newton’s Law III:
Action = Reaction
-
If an external force , F
1
, is applied to an object (
“action”
)
-

object pushes back with a force, F
2
, that is equal in magnitude and opposite in direction to F
1
(
“reaction”
)
ie.
force is always mutual
(principle that makes reaction engines ( jets and rockets ) work)
eg. 1 reaction engine (jet engine, rocket engine)
engine expels gas through nozzle with force, F , in reverse direction
-

expelled gas applies force, F , equal in magnitude, to engine in forward direction
-
-
eg. 2 a car accelerating forward
tires apply force , F , to road in reverse direction
- evidence: loose gravel shoots out from under tires in reverse direction
-

by Newton III : road applies force, F , equal in magnitude, to car in forward
-
direction
-

car accelerates forward ( Newton II: a=F/m )
-
Significance for planets:
- Planet pulls on Sun with same amount of force with which Sun pulls on planet
-

Why does planet orbit around Sun instead of Sun orbiting around planet?

- apply Newton II :
- acceleration of planet , a p
=F/m p
, where m p
= mass of planet
- acceleration of Sun , a s
=F/m s
, where m s
= mass of Sun
- compare mass of Sun and planet: m s
>> m p
-
 a s
<< a p

planet accelerated more by force, F, than Sun is
-
 Sun is the primary and planet is the satellite
-
57
Newton’s Law of Universal Gravitation (“The Law of Gravity”):
the force , F, between the Sun and planets
-
gravity: an attractive force between any two objects that have mass
F g
= GM
1
M
2
/r
2
where: F g
= force due to gravity in N
M
1
, M
2
= mass of objects 1 and 2 in kg
r = distance between centers of objects 1 and 2 in m
(Only in case you are interested: G = Gravitational Constant
-
-
= 6.67

10 -11 N m 2 / kg 2 )
Units: M
’s in
Kg , r in m

F in N
-
-
-
Direction of F g
:
- F is a vector quantity
- F on object 1 = F g
, direction: along straight line toward center of object 2
- F on object 2 = F g
, direction: along straight line toward center of object 1 -
-
-
Mutuality of F g
:
- object 1’s gravitational force on object 2 is the same in magnitude and opposite in
-
-
-
direction to object 2’s gravitational force on object 1
- consistent with Newton Law III :

58
-
the Law of Gravity is an Inverse Square Law
-

1/r 2 factor: if distance between objects, r , doubles , F g
decreases by a factor of
four
-
-
-
Universality of Law of Gravity:
- the Law of Gravity is the same everywhere , on all mass and distance scales
- eg.
force of gravity between a baseball and Earth follows the same law as force
of gravity between Earth and the Sun -
-
-
Significance for planets :
The Sun (mass= M s
) and a planet (mass= M p
) at a distance r p
from the Sun, pull on each other with a force of gravity , F g
, equal to GM s
M p
/r p
2
 ie.
planets constantly falling toward Sun
Why do planets not fall into the Sun due to gravity?
an object can have a velocity , v , directed perpendicular ( ie.
at a right angle) to the force, F, acting on it ( v

F )
-
 ie.
a v directed at a right angle to its acceleration, a ( v
 a )
-
-
-
 trajectory of accelerating object depends on value of v
 a
- if v=0
 trajectory is a straight line in direction of F
- eg.
a dropped object falling straight down
-
- if v << a  trajectory is a parabola
- eg. trajectory of baseball that’s been thrown with v parallel to Earth’s
-
-
surface
- if v

a
 trajectory is an ellipse
 ie. object
-
Kepler’s Laws of orbital motion orbits source of gravity following
- eg. trajectory of satellite orbiting Earth, or planet orbiting Sun
- if v >> a
 trajectory is hyperbola ( escape trajectory)
-
-
- eg. trajectory of space craft leaving vicinity of Earth
- Escape velocity, v esc
: value of v
 a needed to overcome gravity of nearby
object and leave orbit
- v esc
= (2GM/R)
1/2
, where: M = mass of object being escaped from
-
R = radius of object being escaped from
G = Newton’s gravitational constant (from
law of gravity)

59
TIDES
a phenomenon due to the force of gravity
ocean tides on Earth: periodic change in ocean level with period of 12 hours (half a day)
-
-
force of gravity is mutual ( ie.
obeys Newton’s Law III):

Earth and Moon pull on each other with same force of gravity , F g
-
force of gravity is an inverse square law :

side of Earth facing Moon pulled more toward Moon than center of Earth, and center of Earth pulled more toward Moon than side facing away from Moon
-

Earth is stretched by Moon’s gravity along direction toward Moon
-

two bulges in Earth: one toward Moon and one away from Moon
water deforms more than solid rock
-

Earth’s oceans bulge more than solid surface does
as Earth rotates

regions on surface carried through ocean bulges
-

local regions experience two high tides and two low tides each day
-
if Moon near meridian or almost opposite meridian

observer in ocean bulge
 high tide if Moon near Eastern or Western horizons

observer between ocean bulges
 low tide
Spring Tides and Neap Tides:
Sun also pulls on Earth with force of gravity
-
-
-
Spring Tides:
-
- at New Moon and Full Moon
-

Sun, Moon, and Earth aligned

Sun and Moon pull along same direction
-

oceans bulge more

tidal variation more extreme
- note: “spring” a misnomer; spring tides occur in every month of the year
-
Neap Tides:
- at 1 st and 3 rd Quarter Moons
-

direction to Sun and Moon at right angles

Sun and Moon pull along
perpendicular directions
-

oceans bulge less

tidal variation less extreme

60
Tidal synchronization of Moon’s rotation:
Moon experiences tidal force due to Earth’s gravity
-

Moon bulges toward and away from Earth
-

squeezing solid rock along changing direction produces friction
-
 Moon’s rotation has slowed to keep bulges pointing along same direction through
Moon
-

only achieved if P rot
= P orb
-
 ie.
tidal force between Earth and Moon has broken Moon’s rotation rate until
it is synchronized with its revolution rate
-
Moon’s tidal force on Earth is also slowly breaking Earth’s rotation rate
-

Earth’s P rot will eventually become synchronized with P orb
of Moon