Lectures 4 - 6
WEEK 2

LEARNING GOALS
1. Understand the difference between geocentric and heliocentric cosmogonies. Understand the Ptolemaic system and how the Copernican heliocentric system better explains our observations of the Moon and planets.
2. Know how Tycho Brahe revolutionized the practice of astronomy. Know Kepler's three laws and be able to explain them. Understand how Galileo's telescopic observations supported a heliocentric cosmogony.
3. Know Newton's three laws of motion and be able to give examples of each. Know Newton's universal law of gravitation. Be able to explain how Newton used his laws of motion and gravity to obtain Kepler's laws.

been practised by
most ancient civilisations.
Greeks (400 BC to the present)

Explains
motion of stars
rotate with but also drift
slowly with respect to stars.

Explains
motion of stars
Sun & Moon rotate with celestial sphere, but also drift slowly with respect to stars.

retrograde motion of the planets ( wanderers )
In the geocentric view this required epicycles

MARS

•
Earth does not feel as if it’s moving
• Natural state for any body is to be stationary
•
The circle: the perfect form
•
Cycles & epicycles required

Refined the geocentric model to a high degree
•
Very accurate, but also very complicated - 80 circles!
•
Refinements kept being added to account for data.
•
No coherent theory behind it.

Ptolemey’s
13 -Volume
Almagest covered elements of spherical astronomy, solar, lunar, and planetary theory, eclipses, and the fixed stars.
It remained the definitive authority on its subject for nearly 1500 years.

Nicolaus Copernicus (1473 - 1543)
Polish Polymath: Lawyer, physician, economist, canon of the church, and artist.
Gifted in Mathematics and influenced by the ideas of Aristarchus, he turned to
Astronomy in the early 1500’s.

Nicolaus Copernicus (1473 - 1543)
The heliocentric model explains retrograde motion easily.

Nicolaus Copernicus (1473 - 1543)
Worked out many details:
Ordering of planetary orbits.
•
Mercury & Venus, Inferior planets, always seen near Sun.
•
Mars, Jupiter, Saturn, Superior planets, sometimes seen on opposite side of the celestial sphere to Sun, high above horizon - Earth between Sun and these planets.

Nicolaus Copernicus (1473 - 1543)
Explained why planets appear in different parts of the sky on different dates
•
Mercury & Venus, Inferior planets, seen in west near Sunset, then in east just before sunrise - elongation.
•
Mars, Jupiter, Saturn, Superior planets, best seen at night in opposition.

•
Conjunction:
The Earth, Sun and a Planet form a straight line in the direction of the Sun (as seen from the
Earth)
•
Opposition:
The Earth, Sun and a Planet form a straight line in the direction away from the Sun (as seen from the Earth,

•
Inferior Planets:
Inferior planets can never be in opposition (they are cannot be away from the sun as seen from the earth).
•
Two Types of Conjunction:
Inferior conjunction (same side as the earth)
Superior conjunction (opposite side)

•
Elongation of a Planet
Elongation is the angular distance of an inferior planet from the Sun as seen from the earth.

•
Elongation of Inferior Planets:
Greatest Elongation is the maximum angular distance of an inferior planet from the Sun.
Mercury 18 o – 28 o
Venus45 o – 47 o (eliptical orbits)
If visible in the morning: (Eastern Elongation)
If visible in the evening: (Western Elongation)
Minimum Elongation occurs at …….?

•
Elongation of Inferior Planets:
Greatest Elongation is the maximum angular distance of an inferior planet from the Sun.
Mercury 18 o – 28 o
Venus45 o – 47 o (eliptical orbits)
If visible in the morning: (Eastern Elongation)
If visible in the evening: (Western Elongation)
Minimum Elongation occurs at conjunction (0 o either inferior or superior)

•
Elongation of Superior Planets:
The minimum elongation of a superior planet occurs at conjunction (= zero degrees)
The greatest elongation of a superior planet occurs at opposition ( = 180 o )

Elongation Period
• Greatest elongations of a planet happen periodically, with a eastern followed by western, and vice versa .
• The period depends on the relative angular velocity of Earth and the planet, as seen from the Sun.
• The time it takes to complete this period is the synodic period of the planet.

Elongation Period
Let
T be the period between successive greatest elongations,
ω be the relative angular velocity,
ω e
Earth's angular velocity and
ω p the planet's angular velocity.
Then

T

2


Elongation Period
Hence

T

2


Elongation Period
But
ω = ω p
– ω e
Hence

T

2


Elongation Period
But
ω = ω p
– ω e
Hence
Hence T

 p
2

  e

Since
Elongation Period
 
2

T
Hence
Then T

2

T p
2


2

T e
T p/e are the siderial periods

Since
Then
Elongation Period
 
2

T
Hence
T

2

T p
2


2

T e

T e
T p
T e

1
T p/e are the siderial periods

Since
Then
Elongation Period
 
2

T
Hence
T

2

T p
2


2

T e

T e
T p
T e

1
T earth
= 365 days: T venus
= 225 days: T = 584 days

Relationship between synodic and siderial periods
• Copernicus devised a mathematical formula to calculate a planet's sidereal period from its synodic period.

Relationship between synodic and siderial periods
• Copernicus devised a mathematical formula to calculate a planet's sidereal period from its synodic period.
• E = siderial period of the Earth
• P = siderial period of the Planet
• S = the synodic period.

Relationship between synodic and siderial periods
• During the time
S , the Earth moves over an angle of (360°/ E ) S
(assuming a circular orbit) and the planet moves (360°/ P ) S .

Relationship between synodic and siderial periods
• Let us consider an inferior planet. which will complete one revolution before the earth by the time the two return to the same position relative to the sun.

Relationship between synodic and siderial periods
S
P
360

S
E
360

360

Relationship between synodic and siderial periods
S
P
360

S
E
360

360

360 S

SP

360 P
E

S

SP

P
E

1

P

E
P
S

1
P

1
E

1
S

Relationship between synodic and siderial periods
S
P
360

S
E
360

360

360 S

SP

360 P
E

S

SP

P
E

1

P

E
P
S

1
P

1
E

1
S
: For Superior Planets
1
P

1
E

1
S

overtakings of one planet by the other.
for superior planet just swap:
E ↔P
1
P
1
E
1
E
1
P
1
S
1
S for superior planet
Box 4-1

Nicolaus Copernicus (1473 - 1543)
Determined planetary distances from
Sun by geometry in terms 1 AU
Planet------Copernicus---Modern
Mercury 0.38 AU 0.39 AU
Venus 0.72 AU 0.72 AU
Mars 1.52 AU 1.52.AU
Jupiter 5.22 AU 5.20 AU
Saturn 9.07 AU 9.54 AU

Nicolaus Copernicus (1473 - 1543)
•
His results showed that the larger the orbit, the longer the period & the smaller the speed.
•
Noticed variable speed on orbits and so included epicycles to keep using circular motion!
•
This made his model no better than
Ptolemy’s geocentric one to astronomers at the time. MORE EVIDENCE NEEDED

Copernicus’
De Revolutionibus
Orbium Coelestium
(1543, year of his death)
On the Revolutions of the Celestial
Spheres

(1546 - 1601)
Danish Astronomer:
Observed Supernova Nov. 11, 1572
Danish king financed observatory
Uraniborg (sky castle) on Hven Island.
Made measurements of stars and planets with unprecedented accuracy.
Repeated measurements with different instruments to assess errors - pioneer of our modern practices.

(1546 - 1601)
Danish Astronomer:
Observed Supernova Nov. 11, 1572
Danish king financed observatory
Uraniborg (sky castle) on Hven Island.
Made measurements of stars and planets with unprecedented accuracy.
Repeated measurements with different instruments to assess errors - pioneer of our modern practices.

Tycho Brahe (1546 - 1601)
• Attempted to test Copernicus’s ideas about the planets orbiting the Sun.
•
Failed to measure any stellar parallax; concluded Earth was stationary and
Copernicus wrong. (We now know the stars were too far away to measure parallax without a telescope)
•
Compiled a massive data base with
1

= 1 arcmin accuracy
(best one can do without a telescope)

Tycho Brahe (1546 - 1601)
• Attempted to test Copernicus’s ideas about the planets orbiting the Sun.
•
Failed to measure any stellar parallax; concluded Earth was stationary and
Copernicus wrong. (We now know the stars were too far away to measure parallax without a telescope)
•
Compiled a massive data base with
1

= 1 arcmin accuracy
(best one can do without a telescope)

Johannes Kepler (1571 - 1630)
Employed by Tycho in 1600 in Prague.
After Tycho’s death Kepler inherited his data and his position as
Imperial Mathematician of the
Holy Roman Empire.

Johannes Kepler (1571 - 1630)
Employed by Tycho in 1600 in Prague.
After Tycho’s death Kepler inherited his data and his position as
Imperial Mathematician of the
Holy Roman Empire.

Johannes Kepler (1571 - 1630)
Kepler could be said to be the first astrophysicist
He could also be said to be the last scientific astrologer.
(except maybe me)

Johannes Kepler (1571 - 1630)
Astrology was once kind of scientific

Johannes Kepler (1571 - 1630)
Astrology was once kind of scientific
What happened last time Venus rose in the constellation of the goat? Maybe something like it will happen again.

Johannes Kepler (1571 - 1630)
Astrology
Disaster:

Johannes Kepler (1571 - 1630)
Astrology
Disaster: from the Greek for bad star

Johannes Kepler (1571 - 1630)
Astrology
Disaster: from the Greek for bad star
Influenza:

Johannes Kepler (1571 - 1630)
Astrology
Disaster: from the Greek for bad star
Influenza: the influence of the stars

Johannes Kepler (1571 - 1630)
Astrology
Even today, how many papers have a regular astrology column?

Johannes Kepler (1571 - 1630)
Astrology
Even today, how many papers have a regular astrology column?
But how many have a regular astronomy column?

Johannes Kepler (1571 - 1630)
Astrology
Based on the idea that the position of the planets in the sky fundamentally affect our lifes.
But there are greater influences.

Johannes Kepler (1571 - 1630)
Kepler believed in the heliocentric model.
29 years of struggle with the data led him to try elliptical orbits with dramatic success.
He confirmed this by mapping out the shape of orbits by observations with Earth’s orbit (1 AU) as baseline.

Johannes Kepler (1571 - 1630)
In Kepler’s time there were only 6 known planets:
Mercury, Venus, Earth, Mars, Jupiter and Saturn.

Johannes Kepler (1571 - 1630)
In Kepler’s time there were only 6 known planets:
Mercury, Venus, Earth, Mars, Jupiter and Saturn.
Why not 20, or 100?
Why these particular spacings?
Before Kepler no one had asked such questions.

Johannes Kepler (1571 - 1630)
Consider an equilateral triangle,
Draw a circle outside and one inside

Johannes Kepler (1571 - 1630)
Consider an equilateral triangle,
Draw one circle outside, one inside and remove the triangle.

Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the orbit of Jupiter to the orbit of Saturn.

Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the orbit of Jupiter to the orbit of Saturn.
Spooky eh!

Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the orbit of Jupiter to the orbit of Saturn.
Spooky eh! But Kepler was intrigue and expanded on it.

Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the orbit of Jupiter to the orbit of Saturn.
Spooky eh! But Kepler was intrigue and expanded on it. A triangular prism is a tetrahedron

Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the orbit of Jupiter to the orbit of Saturn.
Spooky eh! But Kepler was intrigue and expanded on it. A triangular prism is a tetrahedron

Johannes Kepler (1571 - 1630)
Could a similar geometry relate the orbits of the other planets?

Johannes Kepler (1571 - 1630)
Could a similar geometry relate the orbits of the other planets?
Kepler recalled the regular solids of Pythagoras.
There were five.

Johannes Kepler (1571 - 1630)
Could a similar geometry relate the orbits of the other planets?
Kepler recalled the regular solids of Pythagoras.
There were five.

Johannes Kepler (1571 - 1630)
He believed they nested one within another.
Hence the invisible supports of the 5 solids was the spheres of the 6 planets.

Spheres enclosing solids

Spheres enclosing solids

Spheres enclosing solids
All this, is an attempt to fit the orbits of the planets with harmonics in music.

Johannes Kepler (1571 - 1630)
But no matter how he tried, he could not make it work very well.

Johannes Kepler (1571 - 1630)
But no matter how he tried, he could not make it work very well.
Why not?

Johannes Kepler (1571 - 1630)
But no matter how he tried, he could not make it work very well.
Why not?
Because it was wrong.

Johannes Kepler (1571 - 1630)
But no matter how he tried, he could not make it work very well.
Why not?
Because it was wrong.
The later discovery of Uranus, Neptune, Pluto, and the others prove that

Johannes Kepler (1571 - 1630)
He spent 29 years trying to make it work, but in the end decided that it was the observations that were right, not his ideas.
Hence, he finally abandoned them.
Astronomy wins over astrology

Johannes Kepler (1571 - 1630)
In abandoning his regular solids, he was also able to free his mind of the perfect sphere/circle for orbital motion.
Hence he considered that they may be elliptical.

Johannes Kepler (1571 - 1630)
Kepler’s 3 Laws of planetary motion:
1) Orbital paths of planets are ellipses, with the Sun at one focus.(1609)
2) Line joining the planet to the Sun sweeps out equal areas in equal times.
3) The square of a planet’s orbital period is proportional to the cube of its semimajor axis

Kepler’s 1 st Law
• The orbit of every planet is an ellipse with the Sun at one focus.
Planet
P
Sun at a focus
Empty focus

Kepler’s 1 st Law r


1
 e cos
 r and
 are polar coordinates e is the eccentricity of the ellipse
 is the semi-latus rectum
Planet
P
Sun at a focus
Empty focus


Planet
Kepler’s 1 st Law r and
 are polar coordinates r r
P
Major axis

Eccentricity e
Planet
Kepler’s 1 st Law e
 a
2  b
2 a
2
 r r
P

Semi Major axis a
Semi Minor Axis b
1
 a b
 2

Eccentricity e
Kepler’s 1 st Law e
 a
2  b
2 a
2

1
 a b
 2


Planet
Kepler’s 1 st Law
Semi Latus Rectum
 = b 2 / a r r
P r


1
 e cos

Note that a circle is a special type is ellipse (one with e = 0)

Kepler’s 2 nd Law
The line between the sun and a planet sweeps out equal areas in equal time.

Kepler’s 2 nd Law
The line between the sun and a planet sweeps out equal areas in equal time.
If the planet moves from A to B in one day.
Then the Sun A and B roughly form a triangle.
The area of that triangle is the same no matter where the planet is on its orbit.

Kepler’s 2 nd Law
The orbit is an ellipse.
Thus, the planet must move faster when near perihelion than it does near aphelion.

Kepler’s 2 nd Law
The orbit is an ellipse.
Thus, the planet must move faster when near perihelion than it does near aphelion.
This is because the net tangential force involved in an elliptical orbit is zero.
As the areal velocity is proportional to angular momentum, Kepler's second law is a statement of the law of conservation of angular momentum.
.

Kepler’s 2 nd Law
Written symbolically, d dt

1
2 r
2
 


0
1
2 r
2
  is the " areal velocity"

Kepler’s 3 rd Law
The square of the orbital period of a planet is proportional to the cube of its semi-major axis.
P 2
 a 3

Kepler’s 3 rd Law
The square of the orbital period of a planet is proportional to the cube of its semi-major axis.
P 2
 a 3
Example
Uranus was found to have a period of 84 years.
What is its distance from the Sun?

Kepler’s 3 rd Law
The square of the orbital period of a planet is proportional to the cube of its semi-major axis.
P 2
 a 3
Example
Uranus was found to have a period of 84 years.
What is its distance from the Sun?
a = P 2/3 = 84 2/3 = 19 AU

Using his laws Kepler was the first astronomer to predict a transit of Venus (for the year 1631)

Galileo Galilei (1564 - 1642)
From 1610 onwards he saw: mountains on the Moon, sunspots on the Sun, the rings of Saturn,
Jupiter’s moons ( providing a counter example to the view that Earth is the centre of the universe)

Galileo Galilei (1564 - 1642)
One of the first to use a telescope,
His observations constitute the beginnings of modern astronomy. His defence of the Copernican heliocentric solar system was published in
The Starry Messenger.
(Siderius Nuncius)

Galileo Galilei (1564 - 1642)
One of the first to use a telescope,
His observations constitute the beginnings of modern astronomy. His defence of the Copernican heliocentric solar system was published in
The Starry Messenger.
(Siderius Nuncius)

Galileo Galilei (1564 - 1642)
He noted that as the phases of Venus changed, so did its apparent size.
This provided decisive evidence against
Ptolemaic geocentric system.

Phases of Venus as it orbits a
= angular diameter
(arcsec)

Venus in the Heliocentric system

Venus in the Geocentric system

Galileo Galilei (1564 - 1642)
Jupiter
(the Galilean satellites)
This provided a counterexample to the view that Earth is the centre of the universe

Jupiter’s moons

Isaac Newton (1642 - 1727)
One of the greatest scientists who ever lived: was a great experimentalist, mathematician,
& philosopher of the scientific method
.

Isaac Newton (1642 - 1727)
One of the greatest scientists who ever lived: was a great experimentalist, mathematician,
& philosopher of the scientific method
.

Isaac Newton (1642 - 1727)
Principia Mathematica 1667
Newton’s Laws of Motion:
1) A particle will continue moving in a straight line unless acted on by a force.
2) Application of a force, F causes an acceleration, a , given by ma = F
3) Action & reaction are equal and opposite.

Principia 1667
Newton’s derivation of Centripetal
Acceleration for motion in a circle using:
1) A particle will continue moving in a straight line unless acted on by a force.
2) Application of a force, F causes an acceleration, a , given by ma = F

Position
Centripental
Velocity

Centripental
Position
Draw a position vector
Velocity

Centripental
Position
Draw a position vector
Velocity r

Centripental
Position
Draw a position vector
Velocity v r

Centripental
Position
Draw a position vector
Velocity
Draw that velocity vector v r

Centripental
Position
Draw a position vector
Velocity
Draw that velocity vector v r

Position
Centripental
Velocity
Draw that velocity vector v r
Draw a position vector some time d t later

Position
Centripental
Velocity
Draw that velocity vector r r v
Draw a radius vector some time d t later

Position
Centripental
Velocity v r r v
Draw a position vector some time d t later

Position
Centripental
Velocity v r r v
Draw a position vector some time d t later
Draw that new velocity vector

Position
Centripental
Velocity v r r v
Draw a position vector some time d t later
Draw that new velocity vector

Position
Centripental
Velocity v r r v
Now draw an acceleration vector

Position
Centripental
Velocity v r r v
Now draw an acceleration vector

Position
Centripental
Velocity v r r v
And here
Now draw an acceleration vector

Position
Centripental
Velocity v r r v

Position
Centripental
Velocity v r r v
The time taken for both the position vector and the velocity vector to complete one cycle must be the same.

How long does it take the position to complete one cycle? v r r v

How long does it take the position to complete one cycle?
Circumference divided by the velocity. v r r v

How long does it take the position to complete one cycle?
Circumference divided by the thing that is changing: v .
P

2
 r v v v r r

How long does it take the velocity to complete one cycle?
P

2
 r v v r r v

How long does it take the velocity to complete one cycle?
The circumference divided by the thing that is changing: a
P

2
 r v v v r r

How long does it take the velocity to complete one cycle?
The circumference divided by the thing that is changing: a
P

2
 r v v
P

2
 v a r v r

But the periods P are the same for both.
P

2
 r v v r r v
P

2
 v a

But the periods P are the same for both. Hence,
2
 r

2 v
 v a v v r r

But the periods P are the same for both. Hence,
2
 r v

2
 v a
 a
 v
2 r v v r r

v

 d d dt x x
 v d t
Centripetal Acceleration v r d x d r d d
 d x
 v r d t r v’

d  v d t r d  v d
 d v v v
 d v d t
 v
2 r v d t r
Centripetal Acceleration d v v’ d v v at B v at A

Apply Newton’s 2 nd Law
F
 ma
 m v
2 r

Apply Newton’s 2 nd Law
F
F
 ma
 m v
2 r
  v r
 m

2 r

Isaac Newton (1642 - 1727)
Principia Mathematica 1667
Newton’s Laws of Motion:
1) A particle will continue moving in a straight line unless acted on by a force.
2) Application of a force, F causes an acceleration, a , given by ma=F
3) Action & reaction are equal and opposite.

Principia 1667

Newton’s Law of Universal
Gravitation m
P
M
Sun

Newton’s Law of Universal
Gravitation m
P
M
Sun
F = G
M
S u n m r
2
P

Newton’s Law of Universal
Gravitation m
P
M
Sun
F = G
M
S u n m r
2
P

Newton’s Law of Universal
Gravitation m
P
How did Newton derive this law?
M
Sun
F = G
M
S u n m r
2
P

Newton’s Law of Universal
Gravitation m
P
He made it up
M
Sun
F = G
M
S u n m r
2
P

Newton’s Law of Universal
Gravitation m
P
Its an educated guess
M
Sun
F = G
M
S u n m r
2
P

Newton’s Law of Universal
Gravitation m
P
He made a few educated guesses
Until he found one that worked.
M
Sun
F = G
M
S u n m r
2
P

To keep the planet in an orbit of radius r , requires a centripetal force F (centripetal) .
This is provided by the Sun’s gravitational force F (grav) .
F
= F

Using the astronomer’s notation,
r = a = semi-major axis
Notice that this law applies to all planets, asteroids etc orbiting the sun.
P
2
=




4
π
2
G M sun



 a
3

P = period
a = semi-major axis
M
Sun
= Solar mass (

M
⊙
)
Notice that this law applies to all objects orbiting the sun.
P
2
4
2
GM
Sun a
3
Earth has
P = 1 yr, a = 1 AU
P
2
(yrs) = a
3
(AU)

Kepler’s 2 nd Law

Kepler’s 2 nd Law
The line joining a planet to the sun sweeps out equal areas in equal time.
A consequence of the law of conservation of momentum

Angular Momentum is
L = Momentum
 lever arm
Illustrate for circular motion:
L = mvr = mr

2
L =
m

r
A r v

Area swept out on one second is:
A

 r
2
P r
A r v

Area swept out on one second is: but P = 2p/w
A

 r
2
P r
A r v

Area swept out on one second is: but P =
2/
A

 r
2
 r
P
2

2 v r
A r

Area swept out on one second is: but P =
2/ and v =
 r
A

 r
2
 r
P
2

2 v r
A r

Area swept out on one second is: but P =
2/ and v =
 r
A

 r
2
P
 r
2

2
 vr
2 v r
A r

Conservation of Momentum
L
 mvr
 constant

Conservation of Momentum
L
 mvr
 constant
A
 vr
2

Conservation of Momentum
L
 mvr
 constant
A
 vr

2
L
2 m

Conservation of Momentum
L
 mvr
 constant
A
 vr

2
L
2 m
L , 2 , and m are all constant, hence A must be a constant.

1:1
1:2
SUN:Jupiter reflex motion of SUN 12.4 m/s

rd
P
2
=


G

M
4 π
1
+
2
M
2


 a
3

Earth’s Moon 27.32 days 0.055 5.14

Example
• Jupiter’s moon Europa has a period of 3.55 days and its average distance from the planet is 671,000 km. Determine the mass of
Jupiter.

m
J
 m
E

4
 a
3
GP
2

m
J
 m
E

4
 a
3
GP
2
We know 4,

, a , G , and P ; but neither of the two masses, giving one equation with two unknowns.

m
J
 m
E

4
 a
3
GP
2
We know 4,

, a , G , and P ; but neither of the two masses, giving one equation with two unknowns.

m
J
 m
E

4
 a
3
GP
2
We know 4,

, a , G , and P ; but neither of the two masses, giving one equation with two unknowns.
Make the reasonable assumption that the mass of Europa is zero.

m
J
 m
E

4
 a
3
GP
2
We know 4,

, a , G , and P ; but neither of the two masses, giving one equation with two unknowns.
Make the reasonable assumption that the mass of Europa is zero ( i.e
., that m
J
+ m
E
= m
J
).

m
J

4
 a
3
GP
2

m
J

6 .
67

4

10


11

3
6 .
71


3 .
55
10
8

86400

2

1 .
9

10
27 kg

a in AU P in years
M in solar masses
M
≈ a
3
P
2 a
E u r op a
= 6 7 1 × 1 0 6 / 1 .4 9 6 × 1 0 1 1 = 4 .4 9 × 1 0 -3 A U
P
E u rop a
= 3 .5 5 / 3 6 5 .2 5 = 9 .7
× 1 0
-3 y e ars
M
Ju p it e r
= 0 .9 6 2 × 1 0
-3
M
S u n