Astronomy 171
Solar System Astronomy
Prof. Paul Martini

Lecture 2: Astronomical Numbers

Key Ideas

Scientific Notation: Express numbers using powers of 10; Standard Prefixes (kilo-, mega-, etc.)
Metric System of Units: Units of Length, Time, Mass,; Weight vs. Mass
Units of Angles

Big Numbers

Review of Scientific Notation

Compact way of expressing large and small numbers using powers of 10: For example: 1.496 x 10⁸; Mantissa: the significant digits of the number (1.496); Exponent: the power of ten of the number (8)

Examples

Mass of the Sun:: 1,989,000,000,000,000,000,000,000,000,000 kg; 1.989 x 10³⁰ kg
Size of a Hydrogen Atom:: 0.0000000000106 meters; 1.06 x 10^-11 meters
Eliminate the extraneous zeros and concentrate on the significant digits.

The Metric System

Astronomers use the Metric System: Length in Meters; Mass in Kilograms; Time in Seconds
Metric Units are almost global: The United States, Liberia, and Myanmar still use ''English'' Units

Examples

Length: 1 kilometer = 10³ meters (1000 meters); 1 centimeter = 10^-2 meters (1/100th of a meters); 1 millimeter = 10^-3 meters (1/1000th of a meters); 1 micron = 10^-6 meters (short for micrometer)
Time:: 1 nanosecond = 10^-9 second (billionth of a second); 1 Megayear = 10⁶ second (1 Million years); 1 Gigayear = 10⁹ second (1 Billion years)

Units of Length

The basic unit of length is the meter (m)
Traditional definition:: 1 ten-millionth the distance from the North Poole to the Equator of the Earth
Modern definition:: The distance traveled by light in a vacuum in 1/299792458th of a second
Common length units:: meters, kilometers

Astronomical Units of Length

Astronomical Unit (AU):: Mean distance from the Earth to the Sun; 1 AU = 1.496 x 10⁸ kilometers; Used for distances between planets
Light Year (ly):: Distance traveled by light in 1 Year; 1 ly = 9.46 x 10¹² kilometers; Used for distances between stars
Parsec (pc):: 1 pc = 3.26 ly; Distance at which 1 AU subtends an angle of 1 arcsecond

Distance Examples

Size of the Earth: Diameter: 12,756 km; Circumference = 40,074 km; It would take 2.5 weeks driving at 60mph to travel around the Earth (25,000 miles)
Earth-Moon Distance: The Earth is 384,000 km from the Moon; The distance to the Moon is comparable to how far a very reliable car will drive in its entire lifetime (240,000 miles)
Earth-Sun Distance: The Earth is 1 AU or 149,600,000 km from the Sun; This is nearly 400 times the Earth-Moon distance or nearly 12,000 times the diameter of the Earth. If you travel 60mph, it would take you 178 years to reach the Sun.
Distance from the Sun to the Nearest Star: Proxima Centauri is 4.22 ly (267,000 AU) from the Sun

Units of Time

The basic unit of time is the second
Traditional definition:: 1/86,400th of the mean solar day; (60s x 60min/hr x 24hr/day = 86,400 s/day)
Modern definition:: 9,192,631,700 oscillations of a Cesium-133 atomic clock
Common time units:: seconds, minutes, hours, years

Units of Mass

The basic unit of mass is the kilogram (kg)
Traditional definition:: 1 kg = mass of 1 liter of pure water
Modern definition:: 1 kg = mass of the international prototype of the kilogram; A platium-iridium alloy weight kept at the International Bureau of Weights and Measures in Sevres, France

Mass vs. Weight

Mass and Weight are NOT the same!
Mass:: The amount of matter in an object
Weight:: The force of gravity on an object
Of the two, Mass is more fundamental: Mass is the same everywhere; Weight depends on the local gravity

Mass and Weight Units

Mass and Weight have different units
Metric:: Mass measured in kilograms; Weight measured in Newtons (unit of force)
English:: Mass measured in slugs; Weight measured in pounds
Don't erroneously mix pounds and kilograms!

Units of Angles

A Complete circle is divided into 360 degrees
The Babylonians started this convention:: 360 is close to 365, the days in a year; 360 is dividible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180 without using fractions
Start by quartering the circle (90 degrees), then subdividing further using geometry

Subdividing the Degree

Degrees are divided into Minutes of Arc ('):: 1 degree divided into 60 minutes of arc; minute comes from pars minuta prima (first small part); 1 minute = 1/60th of a degree
Minutes are divided into Seconds of Arc (''):: 1 minute divided into 60 seconds of arc; second comes from parte minutae secundae (second small part); 1 second = 1/3600th of a degree (very small!)

Subdividing the Degree (cont'd)

Question: Why 60?
Answer: Those Babylonians (again) ...: 60 is divisible by 2, 3, 4, 5, 6,10, 12, 15, 20, and 30 without using fractions; (100 is only divisible by 2, 4, 5, 10, 20, 25, 50)
The Babylonians subdivided the degree as fractions of 60, for example:: 7 14/60 degrees
Claudius Ptolemy introduced the modern notation:: 7 degrees 14' 00''

Angles and Distance

If the size of an object is known, its distance can be measured by the angle it subtends.
If the angle is small: d = (57.38/theta) AU for theta measured in degrees
d = (206265/theta) AU for theta measured in arcseconds

The Parsec

A parsec is defined as the distance at which 1 AU will subtend 1 arcsecond.
The angular shift of stars over the course of the year is used to measure the distance of nearby stars
Proxima Centauri, at 1.26 pc, moves 1/1.26 or 0.8 arcseconds when the Earth moves 1 AU (6 months)

See A Note about Graphics to learn why some of the graphics shown in the lectures are not reproduced with these notes.