Powers of 10 :

Measurements and Scales



Contents

  • Scales: Big and Small

  • Powers of 10 Notation

  • Some Examples of Scales and "Orders of Magnitude"


    Scales: Big and Small

    In astronomy, as in all of science, we measure quantities and when we make measurements we present our results with numbers. These measurments relate to sizes of objects, durations, energy of observed processes, and temperature (and more ...). Part of understanding what a number - a numerical value based on some measurments - conveys is actually having a "feeling" for how "big" or how "small" a number is in relation to other numbers. In fact, this is how we understand and relate to the world around us in our daily experiences which involve as varied activities as running, driving, working, and even using our language to talk and relate information to others. Sizes of objects and duration of events make sense only in relation to other sizes and other durations which "we have in mind".

    Suppose someone tell you that Mars is about 80 million kilometers from the Earth. What does this number mean? Well, from your daily experiences this number should sound "very large". Part of this has to do with the fact that in our experience - limited here on the surface of the Earth - we don't travel that large of a distance in a day or two. We go 10 or 20 or 30 or ... few hunderd kilometers. So, 78 million kilometers is very large compared to our dialy distances on the Earth. On the other hand, if someone tell you that the size of an atom is about 0.000000000001 kilometers, you would say that "wow, that is so tiny!". And yes compared to our expereinces it is small.

    Notice that "large" and "small" only make sense in relation - or by reference - to other things that we do have a sense of. If you were told that the size of our Galaxy is about:1,000,000,000,000,000,000 kilometers, then, comparatively this dwarfs the 78 million kilometer distance from Earth to Mars ... all of a sudden, Mars feels like the next door neighbor (which, on the astronomical scale of distances it actually is!).

    So, the basic point of all this is that in order for you to interpret numbers in astronomy, in order to understand what someone means when they give you a distance or a time or an energy that seems "incomprehensible", you should have a feeling of scales of distance and time and energy that are present in the universe. For instance, how large is a typical star compared to the size of the city of Denver? How much energy does the Sun put out in one day, compared to ... say, how much electricity we use in one day? These are the kinds of things that allow us to anchor a reference of number to hold on to, and it is only in relation to such an anchor that astronomically measured quantities will begint to make sense to you.

    Powers of 10

    You might have already noticed that the numbers involved in astronomy range over a very large scale of sizes. The smallest sized objects you'll hear about (the electron and the atom )is much much much smaller than the Earth which is in turn itself much much much smaller than the Size of the Galaxy, and the scales continue up like this to the size of the Universe.

    Since these numbers range over such small and such large values we get tired of writing out so many zeros, as was being written in the pervious section. We introduce something called powers of 10 notation. This is just a very simple introduction; you will get more and more experienced with this notation as you study astronomy and as you go along to the secions and examples on scientific notation and algebraic powers.

    Perhaps it is best to illustrate this with an example. Consider the size of our Galaxy again ...:

    This illustrates the basic idea of powers of 10. Each zero represents a power of 10, and if the zero is to the right of a number it represents a power of 10 larger and if a zero is to the left of a number it represents a power of 10 smaller (in this case we use negative powers). So, we can say that:

    Powers of 10 thus allow us to compare sizes (and durations ... see scales below). We can say that the size of our galaxy is "11 orders of magnitude larger than the distance from here to mars and it is 17 orders of magnitude larger than the distance from school to my house". Basically, powers of 10 allow us to talk about degrees of "largness" and "smallness". This might seem confusing at this point, but it will clarify as you see more and more examples - which are abundant in astronomy.

    Some Examples of Scales and Orders of Magnitude

    Here are some examples of the typical quantities that are measured in astronomy. As you can see, distances and sizes, time, and energy are some of the most important measurments that are made, and thus they constitute some of the most relevant scales and orders of magnitude that you should be familiar with. The following tables should give you some feeling of the "typical" numbers that are associated with each measured quantity. You need not worry about the details of each number - just the orders of magnitude.

    Distance and Size Scales

    Object or Distance

     

    Size

    ( km )

    Universe

     

    1023

    Milky Way Galaxy

     

    1018

    Nearest Star

     

    1013

    Size of Solar System

     

    109

    The Sun

     

    106

    The Earth

     

    103

    A Mountain

     

    100

    Humans

     

    10-3

    A Cell

     

    10-8

    An Atom

     

    10-12

    The Nucleus

     

    10-17

     

     

    Time Scales

     

    Duration/Age

     

    Time

    (s)

    Age of The Solar System

     

    1017

    Last Ice Age

     

    1012

    Human Life Time

     

    109

    A Day

     

    104

    A Lecture

     

    103

    A Moment/A Second

     

    100

    (light) ???

     

    10 ?

    Lifetime of a decaying atomic particle

     

    10 ?

     

    Typical Energy Scales

    Energy Source

     

    Energy (joules)

    Big Bang

     

    1068

    Radio galaxy

     

    1055

    Supernova

     

    1043

    Sun's radiation for 1 year

     

    1034

    Volcanic explosion

     

    1019

    H-bomb

     

    1017

    Thunderstorm

     

    1015

    Lightening flash

     

    1010

    Baseball pitch

     

    102

    Hitting keyboard key

     

    10-2

    Hop of a flea

     

    10-7