Astronomy 100
Lecture 2

Prof. Neal Katz

University of Massachusetts, Amherst

Lectures

Table of Contents

Astro 100

Powers of 10 and the Scale of the Universe

Outline

The Scale of the Universe
Powers of 10 and Scientific Notation
The Speed of Light
Angular Sizes

Terms to know

order of magnitude
exponent
scientific notation
angular size
arcminute
arcsecond
Constellation
Asterism

1. The Scale of the Universe: How Big is Astronomy?

Person:	2 meters (~ 6 feet)
Earth radius:	6378 km (~4000 miles)
Average Earth <--> Sun Distance:	1 Astronomical Unit (AU)~ 150,000,000 km
Average Pluto <--> Sun Distance:	39.44 AU ~ 5,900,000,000 km (size of Solar System)
Sun <--> Proxima Centauri (nearest star):	4.2 light year (l.y.) ~ 265,000 AU ~ 42,000,000,000,000 km
Sun <--> Center of Milky Way Galaxy:	25,000 l.y. ~ 1,500,000,000 AU ~ 240,000,000,000,000,000 km
Milky Way <--> Andromeda Galaxy:	2 million l.y.
Size of Observable Universe:	15 billion l.y.

...And that's ONLY THE BEGINNING!

Two ways to express very large (or very small) numbers:

Define NEW UNITS (AU, l.y., etc.); or
Use Powers of 10 and Scientific Notation

2. Powers of 10 and Scientific Notation

The range of sizes, time, temperature, pressure, density, radiation, etc. etc. in astronomy and physics is so huge that some special techniques are required to keep track of it all.

one	1 = 10⁰	1 = 10⁰	one
ten	10 = 10¹	0.1 = 10^-1	tenth
hundred	100 = 10²	0.01 = 10^-2	hundredth
thousand	1000 = 10³	0.001 = 10^-3	thousandth
million	1,000,000 = 10⁶	0.000001 = 10^-6	millionth
billion	1,000,000,000 = 10⁹	0.000000001 = 10^-9	billionth
etc. ... bigger		etc. ... smaller

Names for scales:

Trillion	Tera	(T)	10¹²
Billion	Giga	(G)	10⁹
Million	Mega	(M)	10⁶
Thousand	Kilo	(K)	10³
...
thousandth	milli	(m)	10^-3
millionth	micro	(mu)	10^-6
billionth	nano	(n)	10^-9
trillionth	pico	(p)	10^-12

Examples:

thousand = 1,000: 10³ = 10 x 10 x 10 = 10¹ x 10¹ x 10¹ = 10¹⁺¹⁺¹

million = 1,000,000: 10⁶ = 10³⁺³ = 10³ x 10³; "a million" = "a thousand thousand"

Lengths Expressed as Powers of 10 (VERY round numbers):

atomic nucleus	10^-16 m
Hydrogen atom	10^-10 m
your height	10⁰ m
earth radius	10⁷ m
earth-sun distance	10¹¹ m
solar system	10¹³ m
nearest star	10¹⁶ m ~ one light year (lyr)
sun to galactic center	10²⁰ m
nearest galaxy	10²² m 10⁶ lyr, or 1,000,000 lyr, or "million" lyr

Scientific Notation is just a way of expressing numbers as two parts:

A power of 10 (gives the approximate scale, or "Order of Magnitude"), multiplied by...
A number somewhere between 1 and 10 (gives the precise value)

For Example:: 15 = 1.5 x 10 = 1.5 x 10¹; 100 = 1.0 x 10²; 1500 = 1.5 x 10³; 39466.98 = 3.946698 x 10⁴; 1 = 1.0 x 10⁰; 0.1 = 1. x 10^-1; 0.035 = 3.5 x 10^-2

Can you see that the power of ten tells you how many digits the decimal place should be moved over?

Examples of some astronomical scales in scientific notation:: 1 AU = 150 million km = 150 x 10⁶ km = 1.5 x 10⁸ km; 1 yr = 3.15 x 10⁷ s; 1 l.y. = (3 x 10⁵ km/s) x 3.15 x 10⁷ s = 9.45 x 10¹² km (~ 1.0 x 10¹³ km); 1 Gyr = 10⁹ yr = 3.15 x 10¹⁶ s; Your Age (roughly) = 20 yr = (20) x (3.15 x 10⁷ s) = 6.3 x 10⁸ s; Age of Universe = 20 Gyr = 20 x 10⁹ yr x (3.15 x 10⁷ s/yr) = 6.3 x 10¹⁷ s

3. The Speed of Light and Units

speed = distance / time = meters / seconds = m / s

c = speed of light = 3 x 10⁸ m/s

c is: constant; finite; fastest speed there is

Consequences:

It takes time for light to travel over distance

time = distance / speed

Light travel times:

Distance		Light Travel Times
earth to sun	1.5x10¹¹ m	5x10²s	8.3 minutes
nearest star	~10¹⁶ m	3x10⁷s	1 year
nearest galaxy	~10²² m	3x10¹³s	1 million, or 10⁶ years
"Event Horizon"	2x10²⁶ m	6x10¹⁷s	20 billion, 2x10¹⁰ years, or 20 Gyr

4. Angular Sizes

How big is the Sun? How about the Moon? You know the Sun is physically much larger (400 times), but they appear the same size in the sky, right? This means their angular size is the same. The Sun is 400 times larger than the Moon, but it's 400 times farther away.

An object's angular size is the angle whose point is at your eyeball and whose sides encompass the object. Obviously if you move closer or further from the object, or if the object changes physical size, then its angular size will change.

Angular sizes are measured in degrees (e.g., 360^o = 1 full circle, 90^o = right angle, etc.), with smaller divisions of 60 "arcminutes" (') per degree, and 60 "arcseconds" ('')per arcminute ( 60 x 60 = 3600 arcseconds per degree).

Some examples:

Betelgeuse (largest star) seen from Earth 0.004''

Smallest detail visible from Earth's surface 1''

Smallest detail visible to naked eye 1' = 60''

Sun or Moon seen from Earth 0.5^o = 30'

Your fingernail at arm's length 1^o = 60'
Your fist at arm's length 10^o

Thumb-to-pinky at arm's length 20^o

Horizon to zenith (point overhead) 90^o