PTYS170A1
Evolution of a Habitable Planet
Earth rising over the Moon (Kaguya, JAXA)

Powers of Ten
(chapter 2 in the text)

Scientific Notation
In astronomy and planetary science, we run into numbers that span an enormous range…
For example:
The number of protons in the Sun:
1,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000 protons
The size of a single proton:
0.0000000000001 cm

Scientific Notation
Scientific notation is a universal system that we use to express big and small numbers in terms of powers of ten .
Number: Powers of Ten:
10
100
1,000
10,000
1
= 10 x 10 = 10 2
= 10 x 10 x 10 = 10 3
= 10 x 10 x 10 x 10
100,000
1,000,000
2
3
4
= 10 x 10 x 10 x 10 x 10
= 10 x 10 x 10 x 10 x 10 x 10
= 10
= 10
5
6
To quickly figure out how many powers of ten a big number has, count the number of places after the leading digit.

Scientific Notation
Scientific notation also works for small numbers…
Number: Powers of Ten:
1
0.1
0.01
0.001
0.0001
0
-1
= 1 ÷ 10 ÷ 10
= 1 ÷ 10 ÷ 10 ÷ 10
0.00001
-2
-3
= 1 ÷ 10 ÷ 10 ÷ 10 ÷ 10
= 1 ÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10
= 10
= 10
-4
-5
To quickly figure out how many powers of ten a small number has, count the number of places before the leading digit – including the zero before the decimal.

Powers of ten form the basis for the metric or SI ( Le
Système international d'unités ) systems.

Scientific Notation
Example: The diameter of the Sun is 1,390,000 km. What is that in scientific notation?
1,390,000 kilometers
There are 6 powers of ten here, so we know it’s going to be something x 10 6
What’s left when we take out the 10 6 ?
1.39 x 10 6 kilometers

Example: The size of a nitrogen atom is 0.0000000071 centimeters. What is this in scientific notation?
0.0000000071 centimeters
There are -9 powers of ten here, so we know it’s going to be something x 10 -9
What’s left when we take out the 10 -9 ?
7.1 x 10 -9 centimeters

Scientific Notation
Question: One Astronomical Unit (AU) is 150,000,000 kilometers. What is this in scientific notation?
(a) 15 x 10 7 kilometers
(b) 1.5 x 10 7 kilometers
(c) 1.5 x 10 8 kilometers
(d) 1.5 x 10 9 kilometers

Scientific Notation
Question: a human red blood cell has a diameter of
0.00082 centimeters. What is this in scientific notation?
(a) 8.2 x 10 4 centimeters
(b) 8.2 x 10 3 centimeters
(c) 8.2 x 10 -3 centimeters
(d) 8.2 x 10 -4 centimeters

An Aside on Units
You’ll notice that all of numbers in the previous examples have units attached to them.
For example:
0.00082 centimeters and 150,000,000 kilometers
And sometimes we use abbreviations:
0.00082 cm and 150,000,000 km

Unit Conversion
Sometimes, it is necessary to convert units.
Example: If I have 2 gallons of milk, how many quarts is that? ( hint: 1 gallon = 4 quarts )

Unit Conversion
Sometimes, it is necessary to convert units.
Example: If I have 2 gallons of milk, how many quarts is that? ( hint: 1 galloon = 4 quarts )
2 gallons x conversion factor
= ?
quarts

Unit Conversion
Sometimes, it is necessary to convert units.
Example: If I have 2 gallons of milk, how many quarts is that? ( hint: 1 galloon = 4 quarts )
2 gallons x conversion factor
= ?
quarts
4 quarts
1 gallon or
1 gallon
4 quarts

Unit Conversion
Sometimes, it is necessary to convert units.
Example: If I have 2 gallons of milk, how many quarts is that? ( hint: 1 gallon = 4 quarts )
2 gallons x
4 quarts
1 gallon
= 8 quarts

Unit Conversion
Question: Saturn orbits 10 AU away from the Sun. How far is that in kilometers? Give your answer in scientific notation.
(Hint: 1 Astronomical Unit (AU) is 150,000,000 kilometers)
(a) 1.5 x 10 6 kilometers
(b) 1.5 x 10 7 kilometers
(c) 1.5 x 10 8 kilometers
(d) 1.5 x 10 9 kilometers

Ratios
In addition to just listing absolute numbers, it’s often convenient to express the relative size of two numbers in the form of a ratio , or a fraction.
Example: The cruising speed of a Boeing 747 is 900 kilometers per hour (560 mph). How much faster is a cruising 747 than you driving a car down Speedway, at
60 kilometers per hour (40 mph)?

Ratios
In addition to just listing absolute numbers, it’s often convenient to express the relative size of two numbers in the form of a ratio , or a fraction.
Example: The cruising speed of a Boeing 747 is 900 kilometers per hour (560 mph). How much faster is a cruising 747 than you driving a car down Speedway, at
60 kilometers per hour (40 mph)?
Speed of 747
Speed of car
= ?
900 km/h
60 km/h
= ?

Ratios
Example: The cruising speed of a Boeing 747 is 900 kilometers per hour (560 mph). How much faster is a cruising 747 than you driving a car down Speedway, at
60 kilometers per hour (40 mph)?
900 km/h
60 km/h
= ?
To demonstrate a point, I’m going to use scientific notation…
9
6 10 1
2 km/h km/h
2 – 1 = 1
9
6 x 10 1 km/h
=
=
?
Dividing powers of ten are easy.
The rule is: 10 A /10 B = 10 (A-B)
?

Ratios
Example: The cruising speed of a Boeing 747 is 900 kilometers per hour (560 mph). How much faster is a cruising 747 than you driving a car down Speedway, at
60 kilometers per hour (40 mph)?
9
6 x 10 1 km/h
3
2 x 10 1 km/h
=
=
?
?
Now we just simplify!
1.5 x 10 1 km/h = ?
1

Ratios
Example: The cruising speed of a Boeing 747 is 900 kilometers per hour (560 mph). How much faster is a cruising 747 than you driving a car down Speedway, at
60 kilometers per hour (40 mph)?
9
6 x 10 1 km/h
3
2 x 10 1 km/h
=
=
?
?
Now we just simplify!
1.5 x 10 1 km/h = ?

Some useful identities
When dealing with scientific notation, there are a few useful rules to remember. These might pop up occasionally in this class…
10 A
B
= 10 ( A – B ) 10 A x
B = 10 ( A + B )
( 10 A ) B = 10 ( A x B )

Ratios
Question: The radius of the Earth is ~6.4 x 10 3 km. The radius of the Moon is ~1.6 x 10 3 km. How many times larger is the Earth’s radius than the Moon’s?
(a) The Earth’s radius is ~0.25-times the Moon’s radius.
(b) The Earth’s radius is ~1-times the Moon’s radius.
(c) The Earth’s radius is ~4-times the Moon’s radius.
(d) The Earth’s radius is ~1000-times the Moon’s radius.

Ratios
Question: NASA’s 2013 budget was ~$20 billion. The total requested budget was ~$4 trillion. What fraction of the total budget goes to NASA?
(a) ~0.25 (25%) Health & Human Services
(b) ~0.20 (20%) Defense
(c) ~0.01 (1.0%) Homeland Security
(d) ~0.005 (0.5%) NASA http://en.wikipedia.org/wiki/2013_United_States_federal_budget

Enough with the Math…
…on to the why we need it.

Video: Powers of Ten (EAMES OFFICE LLC) http://youtu.be/0fKBhvDjuy0

We’re going to closely examine the relative sizes and distances of the planets and Moons within our solar system.
To start, we’re going to construct a scale model of the Earth-Moon system.
Image: Moon rising (ISS024E013421)

The Scale of the Earth-Moon System
Diameter of the Earth:
12,740 kilometers
Diameter of the Moon:
3,480 kilometers
Distance from the Earth to the Moon:
384,400 kilometers
We are going to scale everything to the radius of the Earth:
1 Earth Radius (12,740 kilometers) = 1 centimeter
Image: Earth and Moon (Galileo, NASA/JPL)

The Scale of the Earth-Moon System
To start, we are going to first construct a conversion factor to convert between real distances (in kilometers) to our scaled distances (in centimeters).
Diameter of the Earth (in real life): 12,740 kilometers
Diameter of the Earth (in our scaled model): 1 centimeter*
(I’m using an * to denote that this is a scaled unit)
Conversion Factor:
1 cm*
12,740 km

The Scale of the Earth-Moon System
So what are the scaled sizes for the Earth-Moon system?
Diameter of the Earth (in real life): 12,740 kilometers
12,740 km x
1 cm*
12,740 km
=

The Scale of the Earth-Moon System
So what are the scaled sizes for the Earth-Moon system?
Diameter of the Moon (in real life): 3,480 kilometers
3,480 km x
1 cm*
12,740 km
=

The Scale of the Earth-Moon System
So what are the scaled sizes for the Earth-Moon system?
Earth-Moon Distance (in real life) 384,400 km
384,400 km x
1 cm*
12,740 km
=

The Scale of the Earth-Moon System
Diameter of the Earth:
12,740 kilometers = 1 cm*
Diameter of the Moon:
3,480 kilometers = 0.3 cm*
Distance from the Earth to the Moon:
384,400 kilometers = 30.2 cm*
Image: Earth and Moon (Galileo, NASA/JPL)

The Scale of the Earth-Moon System
To get a real sense of the distance between objects in the solar system, it is useful to consider how long it would take to get there…
How long would it take to fly a plane
(going 900 km/h) to the Moon? travel time = distance / speed

The Scale of the Earth-Moon System
To get a real sense of the distance between objects in the solar system, it is useful to consider how long it would take to get there…
How long would it take to fly a plane
(going 900 km/h) to the Moon?
384,400 km
900 km/hour
= 427 hours
427 hours x
1 day
24 day
= 17.8 days

The Apollo astronauts were able to get to the Moon within 4 days.
Their rockets were considerably more powerful than a 747…
Video: Apollo 8 launch (When We Left the Earth) http://youtu.be/y-cv_JJOxGI

The Scale of the Earth-Moon System
To get a real sense of the distance between objects in the solar system, it is useful to consider how long it would take to get there…
How long would it take to fly a radio signal (traveling at the speed of light,
300,000 km/s) to get there?
384,400 km
300,000 km/s
= 1.3 seconds

The Biosphere
The entire biosphere (which is about 50 kilometers thick) would be a thin shell only about 39 micrometers* thick
Image: Endeavour at ISS (ISS015E21944)

Satellite Orbits
The International Space
Station orbits at an altitude of 330 kilometers.
How far is that from the surface in our scaled model?
0.03 centimeters

Satellite Orbits
Geosynchronous satellites (e.g. weather satellites, communication satellites), orbit higher up ( 2.8 cm* ).

The Scale of the Solar System
To really get a sense of the size of the entire solar system, lets extend our scale model of the
Earth-Moon system outward…
Image: The Solar System, to scale(?)

The Scale of the Sun
The Sun has a diameter of 1,391,000 kilometers.
1,391,000 km
1 cm*
12,740 km
In other words, you could fit over 100 earths side-by-side across the Sun.
Images: Multiwavelength Sun, as of 7:20 AM today (SDO) http://sdo.gsfc.nasa.gov/data/

The Scale of the Solar System
Now, for the rest of the planets…
Image: The Solar System, to scale(?)

The Scale of the Solar System
Form groups of ~4 students. I will assign each group a specific planet(s), and give them information on the planet(s) and its Moons.
(1) Figure out the diameter of the planet, and moons, in our scaled system.
Jupiter
(2) Figure out the scaled distance from the center of the planet to each moon (or, in some cases, the rings of the planet).
(3) Draw a simple diagram of your planet, and moons. Make sure each object is the right size, and the Moons/Rings are located at the right distance from the planet.
(4) Determine the scaled distance from the
Sun to the planet…
(Not to scale)

Scale Solar Neighborhood
The closest star to our
Sun is Alpha Centauri, located 4 light years (the distance light travels in 1 year, or 9.5 x 10 12 kilometers) away.
In our scaled model, that would place Alpha
Centauri about 30,000 kilometers away… or roughly at the Moon!
Image: Solar Neighborhood (ESO)

The Asteroid Belt
Most of the asteroids in the solar system are located in the Main
Asteroid Belt , between the orbits of Mars and
Jupiter. How dense is the asteroid belt?

The Asteroid Belt
If you fly through the asteroid belt, will you run the risk of hitting an asteroid?
Video: Asteroid Field Chase (Star Wars Episode V) http://youtu.be/S2CRs8PAhzg

The Asteroid Belt
The average separation between asteroid in the Main
Belt is about 5 million kilometers (~10-times the Earth-Moon distance).
So if you’re flying through the Main Belt, you can pretty much ignore the asteroids.

Even though the distances between asteroids are large, if given enough time, they can still hit things…
Video: The K-T Extinction Event (Armageddon) http://www.youtube.com/watch?v=N3nyn_yZQ98