The metric and British systems of measurement are the two principal ones in use in the world today. In each, there are units for length, mass, and time, which are often called the fundamental units of measurement, since in principle all other quantities of measurement can be expressed in terms of them.
In the British system, used primarily by England, the United States, and the other English-speaking nations for engineering and commercial purposes, the fundamental units are the foot (ft) as the unit of length, the pound (lb) as the unit of weight or force, and the second (s) as the unit of time. Because of the need for a common system of measurement to support worldwide science and technology, the British system is being phased out in favor of an international system of units (known as the SI system) based on the metric system.
The metric system, developed in France in the late eighteenth century, is based on the decimal system and thus is better suited to computation. Consequently, the metric system has been the primary system of measurement for scientists all over the world, including the English-speaking nations. There have been two widely used versions of the metric system: In the MKS version the fundamental unit of length is the meter (m), the unit of mass is the kilogram (kg), and the unit of time is the second. In the CGS version, the unit of length is the centimeter (cm), the unit of mass is the gram (g), and the unit of time is again the second (s). The international system of units employs the MKS version of the metric system. These systems are summarized in Table 1A.1, including the units for force and energy with their symbols in parentheses.
| Quantity | British Unit | CGS Metric Unit | MKS Metric Unit |
|---|---|---|---|
| length | foot (ft) | centimeter (cm) | meter (m) |
| mass | slug (sl) | gram (g) | kilogram (kg) |
| time | second (s) | second (s) | second (s) |
| force | pound (lb) | dyne (dyne) | newton (N) |
| energy | foot-pound (fp) | erg (erg) | joule (J) |
The unit of time, the second, was originally defined as 1/86,400 of the period for one complete rotation of Earth relative to the Sun. There are 86,400 s in one mean solar day. Because Earth's rotation actually varies slightly, a new definition of the second was introduced in 1964 as the time it takes the cesium 133 atom to make 9,192,631,770 vibrations.
In the CGS version of the metric system, the unit of force is the dyne (dyne), and it is the force needed to accelerate 1 g by 1 cm/s2; that is, dyne = g.cm/s2. Force in the MKS system is the newton (N), named after Sir Isaac Newton, and it is the force necessary to accelerate 1 kg by 1 m/s2, or N = kg*m/s2. Since 1 kg equals 1000 g and 1 m is equal to 100 cm, the newton is 100,000 times the dyne in magnitude, or 1 N = 105 dyne.
For units of energy, the CGS system uses the erg (erg), which is the amount of work done by a force of 1 dyne acting through a distance of 1 cm, while in the MKS system the unit is the joule (J). Since a joule is the work done by a force of 1 N acting through a distance of 1 m, the joule is equivalent to 10 million erg, or 1 J = 107 erg.
Although the units in these notes follow astronomical tradition and are, thus, derived with reference to the CGS system, it may be of interest to relate them to the MKS units and to the British units shown in Tables 1A.2 to 1A.3.
| Quantity | Multiply By | To Obtain |
|---|---|---|
| inches | 2.5400 | centimeters |
| feet | 0.3048 | meters |
| miles | 1.6093 | kilometers |
| slugs | 1.459 x 10-2 | grams |
| pounds | 2.248 x 10-6 | dynes |
| foot-pounds | 1.356 x 10-7 | ergs |
| Quantity | Multiply By | To Obtain |
|---|---|---|
| meters | 102 | centimeters |
| kilometers | 105 | centimeters |
| kilograms | 103 | grams |
| newtons | 105 | dynes |
| joules | 107 | ergs |
A number of important physical constants that are used throughout the text are given in CGS units in Table 1A.4.
| Quantity | Symbol | Value in CGS Units |
|---|---|---|
| velocity of light in vacuum | c | 2.998 x 1010 cm/s |
| gravitational constant | G | 6.673 x 10- cm3/g*s2 |
| Planck's constant | h | 6.626 x 10-27 erg*s |
| mass of hydrogen atom | mH | 1.673 x 10-24 g |
| mass of proton | mp | 1.673 x 10-24 g |
| mass of electron | me | 9.109 x 10-28 g |
| mass of neutron | mn | 1.675 x 10-24 g |
Although the metric and British systems of measurement are useful on the scale of our everyday experience, there are many other units of measurement used by astronomers that are more appropriate to the scale of the phenomenon. Tables A1.5 and A1.6 give the conversions from the CGS system to these special units (abbreviation in parentheses).
| Quantity | Multiply By | To Obtain |
|---|---|---|
| centimeters (cm) | 1.000 x 10-8 | angstroms (A) |
| earth radii (Rearth) | 6.371 x 108 | centimeters (cm) |
| solar radii (Rsun) | 6.960 x 1010 | centimeters (cm) |
| astronomical units (AU) | 1.496 x 1013 | centimeters (cm) |
| astronomical units (AU) | 2.149 x 102 | solar radii (Rsun) |
| light years (ly) | 9.461 x 1017 | centimeters (cm) |
| light years (ly) | 1.359 x 107 | solar radii (Rsun) |
| light years (ly) | 6.324 x 104 | astronomical units (AU) |
| parsecs (pc) | 3.086 x 1018 | centimeters (cm) |
| parsecs (pc) | 2.063 x 105 | astronomical units (AU) |
| parsecs (pc) | 3.262 | light years (ly) |
| Quantity | Multiply By | To Obtain |
|---|---|---|
| earth masses (Mearth) | 5.977 x 1027 | grams (g) |
| solar masses (Msun) | 1.989 x 1033 | grams (g) |
| years (y) | 3.156 x 107 | seconds (s) |
| Galactic years | 2.30 x 108 | years (y) |
| electron volts (eV) | 6.242 x 1011 | ergs |
| solar luminosity (Lsun) | 3.82 x 1033 | ergs per second (erg/s) |
Many times in science, especially in astronomy, we are confronted by extremely large or extremely small numbers. For example, the approximate number of stars in our Galaxy is 400 billion, or 400,000,000,000. Numbers of this size make the arithmetical operations of multiplication and division extremely cumbersome. Therefore, it is convenient to use what is known as scientific notation, which involves powers of ten. We begin by explaining exponents and their manipulations.
If n (known as an exponent) is a positive integer, then xn is defined as the number 1 multiplied n times by x; for example for n = 3, x raised to the third power is
By convention any number raised to the zeroth power is equal to 1. If a is the number 10, then we may form a table of powers of 10 with their accepted prefixes and symbols (Table 1A.7.). For example, 1 million light years (ly) is 1 megalight year (Mly).
| Word | Number | Power | Prefix | Symbol |
|---|---|---|---|---|
| trillion | 1,000,000,000,000 | 1012 | tera | T |
| billion | 1,000,000,000 | 109 | giga | G |
| million | 1,000,000 | 106 | mega | M |
| thousand | 1,000 | 103 | kilo | k |
| hundred | 100 | 102 | hecto | h |
| ten | 10 | 101 | deca | da |
| unit | 1 | 100 | ||
| tenth | 0.1 | 10-1 | deci | d |
| hundredth | 0.01 | 10-2 | centi | c |
| thousandth | 0.001 | 10-3 | milli | m |
| millionth | 0.000,001 | 10-6 | micro | m |
| billionth | 0.000,000,001 | 10-9 | nano | n |
| trillionth | 0.000,000,000,001 | 10-12 | pico | p |
There are rules for multiplication, division, raising to a power, or extracting a root for numbers expressed in exponent form:
Multiplication: Multiplication is accomplished by adding exponents. For example, where the dot denotes multiplication,10-3*102*104 = 10-3+2+4 = 103 = 1000.
Division: Division is accomplished by subtracting exponents. For example,104/102 = 104-2 = 102 = 100.
Exponentiation: Raising a number to a power is accomplished by multiplying exponents. For example,(102)5 = 102*5 = 1010 = 10,000,000,000 or
(10-4)2 = 10-4*2 = 10-8 = 0.000,000,01 or
(2 x 102)3 = 23 x 102*3 = 8 x 106 = 8,000,000.
Roots: Extracting a root is also accomplished by multiplying exponents. For example,(106)1/2 = 106*1/2 = 103 = 1000 or
(10-12)1/3 = 10-12*1/3 = 10-4 = 0.0001.
With these rules in mind we can now move to define scientific notation and consider some examples of its use:
Scientific Notation: Scientific notation is the expression of any number as the product of a number between 1 and 10 times a power of 10.
As examples of scientific notation for expressing numbers, consider the following examples:
From these examples a general rule is apparent for determining the numerical value and algebraic sign of the power of 10 to be used in expressing the number in scientific notation:
Determining the Power of Ten: The number of places the decimal point is shifted indicates the numerical value of the power of 10; if the shift is to the left, the algebraic sign is positive, and if the shift is to the right, the algebraic sign is negative.
For example, in the number 86,496 above, the decimal is shifted four places to the left so that the power of 10 is 4 and its algebraic sign is positive, or 8.6496 x 104. For the number 0.0005492 above the decimal is shifted four places to the right so that the power of 10 is also 4 but its algebraic sign is negative, or 5.492 x 10-4.
As mentioned above, scientific notation finds its greatest utility in multiplication or division, such as in Newton's law of gravitation (Section 3.3):
When numbers expressed in scientific notation are to be added or subtracted, then the following basic rule must be applied:
Addition and Subtraction: In addition or subtraction the power of 10 must be the same for all numbers to be added or subtracted, which will also be the appropriate power of 10 of the answer.
For example, let us add the mass of the Sun and Earth, as we might do in Newton's modified form of Kepler's third law (Section 2.4):
It should also be noted that the units for each quantity in the last two examples are manipulated algebraically just as are the numbers. Thus in multiplication and division the units are multiplied and divided in order to obtain resultant units for the answer. And in addition and subtraction, numbers may be added or subtracted if they have the same units. Hence an extremely valuable means of verifying the correctness of a series of algebraic operations is to carry out first the operations with the units only. The result must be the correct units for the desired answer; that is if the desired answer is a force, then the units will not be those of energy.
Mathematical symbols are a magnificent shorthand for science, allowing scientists to express amazingly complex ideas, concepts, and relations in a reasonably simple universal language. Part of learning to read the language of science is to know the meanings of the symbols listed in Table A1.8.
| Symbol | Meaning | Examples |
|---|---|---|
| = | equal | a equals the sum of b and c: a = b + c |
| proportional to | square of the sidereal period is proportional to the cube of the semimajor axis; P2 a3; or apparent brightness is inversely proportional to distance squared: b 1/d2 | |
| very nearly equal to | 3.998 4 | |
| ~ | roughly or approximately equal to | 3.61 ~ 4; or 87.5 ~ 100 |
| > | greater than | a > b, such as 4 > 3 |
| < | less than | a < b, such as 4 < 7 |
In proportionalities the important point is not the equality of things but how some quantity, called a dependent variable, depends on one or several other quantities, called independent variables. To remove the proportionality, we must introduce into the equation a constant of proportionality that incorporates the units of measurement for the quantities involved. As an example, if k and G are appropriate constants of proportionality, then
where G is known as the constant of gravitation.
Often mathematical relations can be formed as simple ratios, such as in measuring the Doppler effect:
where delta lambda is the change in wavelength, lambda the wavelength, v the radial velocity, and c the velocity of light. Thus the equation says that the ratio of the two wavelengths, delta lambda and lambda, is the same as the ratio of the two velocities, v and c.
A circle on a flat plane can be divided into 360 degrees of arc (360o), with each degree further subdivided into 60 minutes of arc (60') and each minute of arc into 60 seconds of arc (60"). Thus there are 2.16 x 104', or 1.296 x 106", in 360o. Another unit of measure for angle is the radian, there being 2*pi, or 6.2831852, radians in 360o. Thus, 1 radian = 57.29578o, or 1 radian = 206,264.8".
A useful relation when dealing with circles, such as one of the principal reference circles on the celestial sphere (Appendix 2), is the length of an arc of the circle. The relation is:
where the units for the arc length are the same as those for the radius of arc. Since there are 2 radians (360o) subtended by the arc of a whole circle, the circumference (C) is
where r is the radius and d is the diameter (d = 2r) of the circle.
In many places in the text, we discuss areas and volumes. Areas are defined as products of lengths, that is, lengths squared, while volume is area times length, or length cubed. The area (A) of a circle in a plane is
Another useful area for astronomical considerations is the surface area of a sphere. The sphere area (S) is
while the volume (V) enclosed by the surface is given by