The Scale of the Cosmos Lecture 2. POWERS OF 10 & SCIENTIFIC NOTATION In science and especially in...

The Scale of the CosmosThe Scale of the Cosmos

Lecture 2Lecture 2

POWERS OF 10 & SCIENTIFIC NOTATION

• In science and especially in astronomy, you have to deal with very large numbers and very small numbers.

• For example, the number of kilometers in a light year is approximately 9,500,000,000,000 (9.5 trillion).

• The diameter of the hydrogen atom is 0.000000013 centimeters (13 billionths).

To manage large numbers and small numbers, professionals make use of powers of 10 and scientific notation.

10,000,000,000 (10 billion) years is the approximate age of the Milky Way Galaxy (MWG). Instead of writing out the large number in expanded form (i.e., 10,000,000,000) it is written as as a power of 10…1010).

10,000,000,000.

1010

12345678910

The number of place values to movethe decimal behind the 1.

12345

100,000.

105

The diameter of the MWG is approximately 100,000 (100 thousand) light years.

The number of stars in the MWG is approximately 100,000,000,000 (100 billion).

100,000,000,000.

1011

1234567891011

The number of place values to movethe decimal behind the 1.

0.000000001

10-9

87654321

Large numbers have a positive exponent when written as a power of 10. Small numbers have a negative exponent when written as a power of 10.

Consider the small number 0.000000001 (1 billionth):

9

100 = One101 = Ten102 = One Hundred103 = One Thousand104 = Ten Thousand105 = One Hundred Thousand106 = One Million107 = Ten Million108 = One Hundred Million109 = One Billion1010 = Ten Billion1011 = One Hundred Billion1012 = One Trillion

100 = One10-1 = One Tenth10-2 = One Hundredth10-3 = One Thousandth10-4 = Ten Thousandth10-5 = One Hundred Thousandth10-6 = One Millionth10-7 = Ten Millionth10-8 = One Hundred Millionth10-9 = One Billionth10-10 = Ten Billionth10-11 = One Hundred Billionth10-12 = One Trillionth

Positive Exponents Negative Exponents

SCIENTIFIC NOTATION

A number is put into scientific notation if it is in the form d x 10n (large number) or d x 10-n (small number) where d is a decimal between 1 and 10 (i.e., 1 ≤ d < 10).

The distance to the Sun is about 150,000,000 km

150,000,000 = 1.5 x 108

12345678

150,000,000.

1.5 x 108

12345678

38,000,000,000.

3.8 x 1010

10 9

7654321

0.000000478

4.78 x 10-7

54321

0.000031

3.1 x 10-5

To multiply or divide powers of ten you:

Add the exponents when you multiplySubtract the exponents when you divide

105 * 107 = 105+7 = 1012 103 * 104 = 103+4 = 107

108 = 108-5

= 103

105

1012 = 1012-3

= 109

103

ARITHMETIC OF NUMBERS IN SCIENTIFIC NOTATION

(2.1 x 103) . (1.5 x 106) = (2.1 . 1.5) x (103 . 106) = 3.15 x 109

Multiply the Decimals Add the Exponents

7.2 x 1017 = 7.2 x 1017 = 2.4 x 107

3 x 1010 3 1010

Divide the Decimals Subtract the Exponents

When it comes to the arithmetic of numbers in scientific notation, let your fingers do the walking…USE YOUR CALCULATOR.

Your calculator will handle all of the decimal calculations and the addition/subtraction of the exponents.

All you have to do is to set your calculator to scientific notation mode then key in the numbers in the arithmetic problem.

Note – DO NOT enter a number as “3” “x” “10” “^4”, it will treat the “3” and the “10^4” as separate numbers. Use the “exp” or “EE” button (for “exponent”) as in “3” “EE” “4”. (Often written in shorthand form as 3e4).

SAMPLE PROBLEMS

Light travels at the speed of c = 300,000 km/s. The distance from Earth to the Sun is 150,000,000 km. How long does it take light travel from the Sun to Earth?

Distance = Speed . Time

Speed = Distance

Time

Time = Distance

Speed

Time = Distance = 150,000,000 km

Speed 300,000 km/s

Time = 500 seconds

500 s . 1 min/60 s = 8.3 minutes

The Earth is 8.3 light-minutes from the Sun.

The metric system as opposed to the British system of units is used in science. There are two versions of the metric system:

MKS = Meters (m) Kilograms (kg) Seconds (s)CGS = Centimeters (cm) Grams (g) Seconds (s)

It is customary to use CGS in stellar astronomy and MKS in the other branches of astronomy. We will be using the MKS system of units.

The meter is the unit of distance in the MKS system. It is approximately one yard in length (39.3 inches).

The kilogram is the unit of mass (weight). It is approximately two pounds of weight at Earth’s surface.

Scales of Size and TimeScales of Size and Time

Astronomy deals with objects on a vast range of size scales and time scales.

Most of these size and time scales are way beyond our every-day experience.

Humans, the Earth, and even the solar system are tiny and unimportant on cosmic scales.

A Campus SceneA Campus Scene

16 x 16 m

(52 x 52 ft)

A City ViewA City View

1.6 x 1.6 km

(1 x 1 mile)

The Landscape of PennsylvaniaThe Landscape of Pennsylvania

160 x 160 km

(100 x 100 miles)

The EarthThe Earth

Diameter of the Earth: 12,756 km

Earth and MoonEarth and Moon

Distance Earth – Moon: 384,000 km

Highlands – Older Surface4.5 billion years old

EARTH’S MOON

Maria – Younger Surface3 billion years old

No atmosphere.

Diameter: 3500 km

Sidereal Period: 27.3 days Synodic Period: 29.5 days

Distance from Earth:385,000 km

Earth Orbiting Around the SunEarth Orbiting Around the Sun

Distance Sun – Earth = 150,000,000 km

Earth Orbiting Around the SunEarth Orbiting Around the SunIn order to avoid large numbers beyond our imagination, we introduce new units:

1 Astronomical Unit (AU) = Distance Sun – Earth = 150 million km

(93 million miles)

SUN

Diameter: 1,400,000 kmRotation Period: 25 daysSurface Temperature: 5,800 KCore Temperature: 15,000,000 K

Boiling Earth-Sized Convection Cells

Sunspots (Magnetic Storms)

Solar Flares & Prominences

The Solar SystemThe Solar System

Approx. 100 AU

MERCURY

With no atmosphere, Mercury is heavily cratered by4.5 billion years of meteoritic impacts. Nearest the Sun of all the planets.

60,000,000 km (0.4 AU)

Diameter: 5000 km

Orbital Period: 88 days

Rotational Period: 58 days

Although not much bigger than Earth’s Moon it is much denser, 5.5 times that of water compared to the Moon’s 3.5 times.

VENUSVenus has a hot thick atmosphere. It is so thick that optical-based telescopes cannot penetrate to the surface. Closer to the Sun than Earth, the temperature at the surface is a blistering 800o F.

Diameter: 12,100 km (almost a match for Earth)

Orbital Period: 225 days

Rotational Period: 243 days (retrograde)

Distance from Sun: 110,000,000 km (0.7 AU)

Thick atmosphere of carbon dioxide and sulfuric acid.

Soviet Venra Spacecraft photograph of the surface of Venus.

MARS

Gigantic gorge in the surface of Mars (Valles Marinaris)stretching 3000 km across. It would reach across the entire continental United States.

Diameter: 6800 km

Orbital Period: 1.88 years

Rotational Period: 24h

Distance from Sun:225,000,000 km (1.5 AU)

Thin atmosphere of carbon dioxide and water.

Polar CapDry Ice & Water Ice

Gaspra Ida

Dactyl

ASTEROID BELT

The Asteroid Belt is located between 2 and 4 AU from the Sun. It contains billions of rock boulders. The two at the right are 20-60 km in size.

JUPITERDiameter: 143,000 km

Orbital Period: 12 years


Distance from Sun: 680,000,000 km (5 AU)

Great Red Spot

Atmospheric cloud bands due tohigh winds.

Jupiter is the largest of the planets in the solar system. It is large enough to fit all of the other planets inside of it…twice! It can easily engulf over 1000 Earths.

It has an extensive atmosphere tens of thousands of kilometers thick. It is believe to have been the first planet to form in the solar system, 100 million years after the Sun formed from a large cloud of gas and dust in the rotational plane of the Milky Way Galaxy.

Extensive atmosphere of methane and ammonia.

THE GALILEAN SATELLITESThis composite includes the four largest moons of Jupiter which are known as the Galilean satellites. From left to right, the moons shown are Ganymede, Callisto, Io, and Europa.

The Galilean satellites were first seen by the Italian astronomer Galileo Galilei in 1610. In order of increasing distance from Jupiter, Io is closest, followed by Europa, Ganymede, and Callisto.

Moon-wide ocean encrusted in ice

Most volcanically-active objectIn the solar system

SATURN

Extensive system of rings

Diameter: 121,000 km


Rotational Period: 10h 30m

Distance from Sun:1,400,000,000 km (9.5 AU)

URANUS NEPTUNEDiameter: 51,000 km



Distance from Sun: 2,900,000,000 km (19 AU)

Diameter: 50,000 km




Atmospheric clouds

OORTCLOUD

&KUIPER

BELT

(a) Diagram of the Oort cloud, showing a few cometary orbits. Most Oort cloud comets never come close to the Sun. Of all the orbits shown, only the most elongated ellipse represents a comet that will actually enter the solar system (which is smaller than the dot at the center of the figure on this scale) and possibly become visible from Earth.

(b)The Kuiper belt, the source of the short-period comets, whose orbits hug the ecliptic plane.

Pluto

Charon

PLUTO

Diameter: 2,200 km


Rotational Period: 6 days


Discovered in the 1930’s by Clyde Tombaugh, Pluto was long considered the 9th planet in the solar system. Although it built up in the same way as the planets, it is now better classified as a Kuiper Belt Object (KBO).

Along with newly-discovered KBO Eris and the asteroid Ceres, it is now also classified as a dwarf planet.

The Kuiper Belt is a region outside Neptune’s orbit where billions of ice boulders (the building blocks of the planets) are located. The second KBO after Pluto was discovered in 1992, although their presence was predicted circa 1950 by Gerard Kuiper.

(Almost) Empty Space Around Our (Almost) Empty Space Around Our Solar SystemSolar System

Approx. 10,000 AU

The Solar NeighborhoodThe Solar Neighborhood

Approx. 17 light years

The Solar NeighborhoodThe Solar Neighborhood

Approx. 17 light years

New distance scale:

1 light year (ly) =

Distance traveled by light in 1 year

= 63,000 AU = 1013 km

= 10,000,000,000,000 km

(= 1 + 13 zeros)

= 10 trillion km

Nearest star to the Sun:

Proxima Centauri, at a distance of 4.2 light years

The Extended Solar NeighborhoodThe Extended Solar Neighborhood

Approx. 1,700 light years

The Milky Way GalaxyThe Milky Way Galaxy

Diameter of the Milky Way: ~ 75,000 ly

The Local Group of GalaxiesThe Local Group of Galaxies

Distance to the nearest large galaxies: several million light years

The Universe on Very Large ScalesThe Universe on Very Large Scales

Clusters of galaxies are grouped into superclusters.

Superclusters form filaments and walls around voids.

For next timeFor next time

Read Units 5 and 6Read Units 5 and 6

Remember that the homework is due on Remember that the homework is due on MondayMonday

The Scale of the Cosmos Lecture 2. POWERS OF 10 & SCIENTIFIC NOTATION In science and especially in...

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