0.1: Math and Measurements

Numbers in Astronomy

Use reassuring language - toolkit includes a lot of calculations and equations - they are there for anyone that wants to see them, you do not have to memorize them or be able to replicate the calculations to understand the rest of the textbook.  Different teachers/classes may adjust the math level

Big numbers,

equations- variables, proportionality

measurements - metric and dimensions

  - light years

Written for classes with no math pre-requisite

Will include equations, example exercises. math problems using algebra?

Next section:

1. velocity m/s
2. acceleration/gravity m/s²

Practice unit conversions - dimension analysis?

how should i format math, with Tex or html?

. First, we use a system for writing large and small numbers called scientific notation. This system eliminates the many zeros of very large or very small numbers. In scientific notation, if you want to write a number such as 500,000,000, you express it as 5 x 10⁸. This also works for very small numbers by using a negative number in the exponent. In this way, 0.00000003 can be represented as 3 x 10^-8.

A common unit astronomers use to describe distances in the universe is a light-year, which is the distance light travels during one year. Even though it has year in its name, it does not measure time. Because light always travels at the same speed, it makes a good standard for keeping track of distances. The speed of light is 3 x 10⁵ km/s and a light year is 9.5 x 10¹² km.

There is another reason the speed of light is such a natural unit of distance for astronomers. Information about the universe comes to us almost exclusively through various forms of light, and all such light travels at the speed of light. This sets a limit on how quickly we can learn about events in the universe. If a star is 100 light-years away, the light we see from it tonight left that star 100 years ago. If anything happened to the star between the time the light left the star and now, we won't know until the event is 100 years old.

What at first may seem a challenge is actually a tremendous benefit in disguise. If astronomers really want to piece together what has happened in the universe since its beginning, they need some kind of data from the past. Fortunately, the farther out in space we look, the longer the light has taken to get here. By looking billions of light-years out into space, astronomers are actually seeing billions of years into the past. In this way, we can reconstruct the history of the cosmos and get a sense of how it has evolved over time.

This is one reason why astronomers strive to build telescopes that can collect more and more of the faint light in the universe. The more light we collect, the fainter the objects we can observe. On average, fainter objects are farther away and can, therefore, tell us about periods of time even deeper in the past. As each new telescope comes online, astronomers are able to see farther into the past.

Scientific Notation

Scientific notation is used to express very large and very small numbers as a product of two numbers. The first number of the product, the digit term, is usually a number not less than 1 and not greater than 10. The second number of the product is the exponential term. An exponential term is written with 2 separate numbers, the base and the exponent. An example with 6 as the base and 3 as the exponent is written 6³. The value of the exponent is the number of times to multiply the base times itself, so 6³=6×6×6=216. In scientific notation the base is always 10. A number in scientific notation with a digit term of 2.45 and an exponential term of \(10^6\) is written \(2.45×10^6\). Some examples of scientific notation are:

\[\begin{align*}
1000&=1×10^3\\
100&=1×10^2\\
10&=1×10^1\\
1&=1×10^0\\
0.1&=1×10^{−1}\\
0.001&=1×10^{−3}\\
2386&=2.386×1000=2.386×10^3\\
0.123&=1.23×0.1=1.23×10^{−1}
\end{align*} \nonumber \]

Scientific notation is also called powers of 10 notation. Note that 10² also designates or 10 × 10, which equals 100. And 9.65 × 100 is just 965. Another way to look at scientific notation is that we separate out the complicated numbers out front, and leave the units of ten to indicate the size of the number. This means that the digit term only ever has one number to the left of the decimal point. So a number like 1,372,568 becomes 1.372568 times a million (10⁶) or \(1.372568×10^6\). The value of the exponent is also the number of times that the decimal point (.) is moved. If we include the decimal point in the above number, it would be 1,372,568. with the decimal point after the 8. We had to move the decimal point six places to the left to get the number into the form where there is only one digit to the left of the decimal point. The reason we call this powers-of-ten notation is that our counting system is based on increases of ten; each place in our numbering system is ten times greater than the place to the right of it.

Measurements in Science

Numerical values for physical quantities are required to study the universe. To comprehend these vast ranges, we must also have accepted units in which to express them. For example, to measure distance and time scientists define specific units to measure them. Measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in units of meters or kilometers. Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way. Units, such as liters, pounds, and centimeters, are standards of comparison for measurements. A 2-liter bottle of a soft drink contains a volume of beverage that is twice that of the accepted volume of 1 liter.

The SI Units

The acronym “SI” is derived from the French Système International. In science, measurements use the metric system,. Its great advantage is that every unit increases by a factor of ten. For example, in the metric system there are 10 millimeters in one centimeter.

For this textbook, we will start with four base units that are listed in Table \(\PageIndex{1}\). These base units are measurements of length, mass, time, and temperature. The standards for these units are fixed by international agreement. SI units have been used by the United States National Institute of Standards and Technology (NIST) since 1964. Units for other properties, such as volume and and density, may be derived from these base units. The base unit of mass is unique in that a decimal prefix, a term defined in the next section, is included. That is, it is the kilogram, not the gram, as you might expect. The base unit of time is the only one that is not metric. Numerous attempts to make it so have never garnered any success; we are still stuck with the 24:60:60 system that we inherited from ancient times.

Table \(\PageIndex{1}\): SI base units

Observable	Base Unit	Abbreviation
length	meter	m
mass	kilogram	kg
time	second	s
temperature (absolute)	Kelvin	K

The SI Decimal Prefixes

Metric systems have the advantage that conversions of units involve only powers of 10, or order of magnitude. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by using an appropriate decimal prefix, listed in Table \(\PageIndex{2}\). For example, distances in meters are suitable in construction, while distances in kilometers are appropriate for air travel, and the tiny measure of nanometers are convenient in optical design. With the metric system there is no need to invent new units for particular applications.

In this textbook, we will be discussing events that occurred a billion years ago, which can also be written 10⁹ years, or 1 Gyr, which is read one giga-year. We will also mention objects that are so small that they must be measured in in nanometers (nm). There is no need to memorize this entire list. Instead use it, or one like it to help understand the scale of a measurement.

Table \(\PageIndex{2}\): SI Decimal Prefixes

Prefix	Symbol	Value	Example (some are approximate)
exa	E	1018	exameter	Em	1018 m	distance light travels in a century
peta	P	1015	petasecond	Ps	1015 s	30 million years
tera	T	1012	terawatt	TW	1012 W	powerful laser output
giga	G	109	gigahertz	GHz	109 H	a microwave frequency
mega	M	106	megacurie	MCi	106 C	high radioactivity
kilo	k	103	kilometer	km	103 m	about 6/10 mile
hecto	h	102	hectoliter	hL	102 L	26 gallons
deka	da	101	dekagram	dag	101 g	teaspoon of butter
—	—	100 (=1)
deci	d	10−1	deciliter	dL	10−1 L	less than half a soda
centi	c	10−2	centimeter	cm	10−2 m	fingertip thickness
milli	m	10−3	millimeter	mm	10−3 m	flea at its shoulders
micro	µ	10−6	micrometer	µm	10−6 m	detail in microscope
nano	n	10−9	nanogram	ng	10−9 g	small speck of dust
pico	p	10−12	picofarad	pF	10−12 F	small capacitor in radio
femto	f	10−15	femtometer	fm	10−15 m	size of a proton
atto	a	10−18	attosecond	as	10−18 s	time light crosses an atom

The Base Units

There are technically seven base units, but to simplify things we are limiting ourselves to the ones we are using in this textbook: length, mass, time, and temperature. If you want to learn about amount of material, in moles, electric current, or luminous intensity, you can visit the NIST website included in the Further Exploration box at the end of this page.

Length

The standard unit of length in both the SI system is the meter (m). A meter was originally specified as 1/10,000,000 of the distance from the North Pole to the equator. It is now defined as the distance light in a vacuum travels in 1/299,792,458 of a second. A meter is about 3 inches longer than a yard (Figure \(\PageIndex{1}\)); one meter is about 39.37 inches or 1.094 yards. Longer distances are often reported in kilometers (1 km = 1000 m = 10³ m), whereas shorter distances can be reported in centimeters (1 cm = 0.01 m = 10⁻² m) or millimeters (1 mm = 0.001 m = 10⁻³ m). To convert from the American system, here are a few helpful factors: 1 mile is 1.61 kilometers (km) and 1 inch is 2.54 centimeters (cm).

Mass

Roughly speaking, mass is a measure of the amount of matter in something. The quantity or amount of matter in an object is determined by the numbers of atoms and molecules of various types it contains. Unlike weight, mass does not vary with location. The mass of an object is the same on Earth, in orbit, or on the surface of the Moon. In practice, it is very difficult to count and identify all of the atoms and molecules in an object, so masses are not often determined in this manner. The SI unit for mass is the kilogram (kg). It is currently defined to be the mass of a platinum-iridium cylinder kept with the old meter standard at the International Bureau of Weights and Measures near Paris. Exact replicas of the standard kilogram are also kept at the United States’ National Institute of Standards and Technology, or NIST, located in Gaithersburg, Maryland outside of Washington D.C., and at other locations around the world. The determination of all other masses can be ultimately traced to a comparison with the standard mass. At the November 2018 meeting of the International Committee for Weights and Measures, a proposed re-definition of kilogram was approved. This proposal (effective May 2019) re-defines the kilogram by precisely fixing the numerical value of Planck's constant (hh) to be exactly 6.62607015 x 10^-34 kg m² s^-1. This is similar to definition of the meter by fixing the speed of light.

A weight of 1 lb is equivalent on Earth to a mass of 0.4536 kg, while a weight of 1 oz is produced by a mass of 28.35 g. Although weight is related to mass, it is not the same thing. Weight refers to the force that gravity exerts on an object. This force is directly proportional to the mass of the object. The weight of an object changes as the force of gravity changes, but its mass does not. An astronaut’s mass does not change just because she goes to the moon. But her weight on the moon is only one-sixth her earth-bound weight because the moon’s gravity is only one-sixth that of the earth’s.

Time

The SI unit for time, the second (s), has a long history. For many years it was defined as 1/86,400 of a mean solar day. More recently, a new standard was adopted to gain greater accuracy and to define the second in terms of a non-varying, or constant, physical phenomenon because the solar day is getting longer due to very gradual slowing of the Earth’s rotation. Cesium atoms can be made to vibrate in a very steady way, and these vibrations can be readily observed and counted using an atomic clock. In 1967 the second was redefined as the time required for 9,192,631,770 of these vibrations. Accuracy in the fundamental units is essential, because all measurements are ultimately expressed in terms of fundamental units and can be no more accurate than are the fundamental units themselves.

Temperature

The concept of temperature has evolved from the common concepts of hot and cold. In science, heat is the quantity of thermal energy that enters or leaves a body. Meanwhile, temperature measures the average translational kinetic energy of the molecules in a body. Since it is challenging to measure the average speed of a molecule, temperature  is operationally defined to be what we measure with a thermometer.

Thermometers are used to measure temperature according to well-defined scales of measurement, which use pre-defined reference points to help compare quantities. A temperature scale can be created by identifying two easily reproducible temperatures, usually the freezing and boiling temperatures of water at standard atmospheric pressure. The SI unit of temperature is the kelvin (K). Note that temperatures in Kelvin are a number an the letter K, without a degree symbol (°), such as 275 K. The degree Celsius (°C) is also allowed in the SI system, with both the word degree and the degree symbol used for Celsius measurements. Celsius degrees are the same magnitude as those of kelvin, but the two scales place their zeros in different places. Water freezes at 273.15 K (0 °C) and boils at 373.15 K (100 °C). All molecular motion ceases at about −459 °F = −273 °C = 0 K, a temperature called absolute zero. Kelvin temperature is measured from this lowest possible temperature, and it is the temperature scale most often used in astronomy All three temperature scales are included in Figure \(\PageIndex{1}\) along with the boiling and freezing points of water and absolute zero.

Converting between Celsius and Fahrenheit is easy if you bear in mind that between the freezing and boiling points of water there are 180 Fahrenheit degrees, but only 100 Celsius degrees, making the F° 100/180 = 5/9 the magnitude of the C°. Because the freezing point is at 32 °F, the two scales are offset by this amount.

The full conversion formulas are:

• K = °C + 273
• °C = 0.555 × (°F – 32)
• °F = (1.8 × °C) + 32

To convert to Kelvin, since 0 °C = 273 K, add 273 to the temperature in Celsius. For example, 25 °C is 298 K.

Properties of Matter

The science of chemistry developed from observations made about the nature and behavior of different kinds of matter, which we refer to collectively as the properties of matter. The properties we refer to in this lesson are all macroscopic properties: those that can be observed in bulk matter. At the microscopic level, matter is of course characterized by its structure: the spatial arrangement of the individual atoms in a molecular unit or an extended solid. By observing a sample of matter and measuring its various properties, we gradually acquire enough information to characterize it; to distinguish it from other kinds of matter. This is the first step in the development of chemical science, in which interest is focused on specific kinds of matter and the transformations between them.

If you think about the various observable properties of matter, it will become apparent that these fall into two classes. Some properties, such as mass and volume, depend on the quantity of matter in the sample we are studying. Clearly, these properties, as important as they may be, cannot by themselves be used to characterize a kind of matter; to say that “water has a mass of 2 kg” is nonsense, although it may be quite true in a particular instance. Properties of this kind are called extensive properties of matter.

Suppose we take further measurements, and find that the same quantity of water whose mass is 2.0 kg also occupies a volume of 2.0 liters. We have measured two extensive properties (mass and volume) of the same sample of matter. This allows us to define a new quantity, the quotient m/V which defines another property of water which we call the density. Unlike the mass and the volume, which by themselves refer only to individual samples of water, the density (mass per unit volume) is a property of all samples of pure water at the same temperature. Density is an example of an intensive property of matter.

This definition of the density illustrates an important general rule: the ratio of two extensive properties is always an intensive property.

Intensive properties are extremely important, because every possible kind of matter possesses a unique set of intensive properties that distinguishes it from every other kind of matter. Some intensive properties can be determined by simple observations: color (absorption spectrum), melting point, density, solubility, acidic or alkaline nature, and density are common examples. Even more fundamental, but less directly observable, is chemical composition.

The more intensive properties we know, the more precisely we can characterize a sample of matter.

Intensive properties are extremely important, because every possible kind of matter possesses a unique set of intensive properties that distinguishes it from every other kind of matter. In other words, intensive properties serve to characterize matter. Many of the intensive properties depend on such variables as the temperature and pressure, but the ways in which these properties change with such variables can themselves be regarded as intensive properties.

To sum up, mass is how much, volume is how big, and density is how tightly packed.

Volume

Volume is the measure of the amount of space occupied by an object. The standard SI unit of volume is defined by the base unit of length. The standard volume is a cubic meter (m³), a cube with an edge length of exactly one meter. To dispense a cubic meter of water, we could build a cubic box with edge lengths of exactly one meter. This box would hold a cubic meter of water or any other substance, as shown in (Figure \(\PageIndex{2}\)).

Pressure

Pressure is the measure of the force exerted on a unit area of surface. Its SI units are therefore newtons per square meter, but we make such frequent use of pressure that a derived SI unit, the pascal, is commonly used:

1 Pa = 1 N m^–2

 

The concept of pressure first developed in connection with studies relating to the atmosphere and vacuum that were first carried out in the 17th century. The molecules of a gas are in a state of constant thermal motion, moving in straight lines until experiencing a collision that exchanges momentum between pairs of molecules and sends them bouncing off in other directions.

The concept of pressure first developed in connection with studies relating to the atmosphere and vacuum that were first carried out in the 17th century. The molecules of a gas are in a state of constant thermal motion, moving in straight lines until experiencing a collision that exchanges momentum between pairs of molecules and sends them bouncing off in other directions. This leads to a completely random distribution of the molecular velocities both in speed and direction— or it would in the absence of the Earth’s gravitational field which exerts a tiny downward force on each molecule, giving motions in that direction a very slight advantage. In an ordinary container this effect is too small to be noticeable, but in a very tall column of air the effect adds up: the molecules in each vertical layer experience more downward-directed hits from those above it. The resulting force is quickly randomized, resulting in an increased pressure in that layer which is then propagated downward into the layers below.

 

The Barometer

In the early 17th century, the Italian physicist and mathematician Evangalisto Torricelli invented a device to measure atmospheric pressure. The Torricellian barometer consists of a vertical glass tube closed at the top and open at the bottom. It is filled with a liquid, traditionally mercury, and is then inverted, with its open end immersed in the container of the same liquid. The liquid level in the tube will fall under its own weight until the downward force is balanced by the vertical force transmitted hydrostatically to the column by the downward force of the atmosphere acting on the liquid surface in the open container. Torricelli was also the first to recognize that the space above the mercury constituted a vacuum, and is credited with being the first to create a vacuum.

 

One standard atmosphere will support a column of mercury that is 76 cm high, so the “millimeter of mercury”, now more commonly known as the torr, has long been a common pressure unit in the sciences: 1 atm = 760 torr.

\[1\, \text{Pa} = 1\, \text{N}\, \text{m}^{–2}\]

Pressure of Earth's Atmosphere

The molecules of gas in Earth's atmosphere are in constant motion. This leads to a completely random distribution of the molecular velocities both in speed and direction— or it would in the absence of the Earth’s gravitational field which exerts a tiny downward force on each molecule, giving motions in that direction a very slight advantage. In an ordinary container this effect is too small to be noticeable, but in a very tall column of air the effect adds up: the molecules in each vertical layer experience more downward-directed hits from those above it. The resulting force is quickly randomized, resulting in an increased pressure in that layer which is then propagated downward into the layers below.  Figure \(\PageIndex{3}\)

 

Add to attributions info = Chem1 Lower.  ccby4[ccby4l]  Stephen Lower

At sea level, the total mass of the sea of air pressing down on each 1-cm² of surface is about 1034 g, or 10340 kg m^–2. The force (weight) that the Earth’s gravitional acceleration g exerts on this mass is

\[\begin{align} f &= ma \nonumber \\[4pt] &= mg \nonumber \\[4pt] &= (10340\text{ kg})(9.81\text{ m s}^{–2}) \nonumber \\[4pt] &= 1.013 \times 10^5\text{ kg m s}^{–2} \nonumber \\[4pt] &= 1.013 \times 10^5 \text{ newtons} \end{align}\]

resulting in a pressure of

\[1.013 \times 10^5\, \text{N}\, \text{m}^{–2} = 1.013 \times 10^5 \text{ Pa}.\]

The actual pressure at sea level varies with atmospheric conditions, so it is customary to define standard atmospheric pressure as 1 atm = 1.01310⁵ pa or 101 kpa. Although the standard atmosphere is not an SI unit, it is still widely employed. In meteorology, the bar, exactly 1.000 × 10⁵ = 0.967 atm, is often used.  Figure \(\PageIndex{4}\)

 

Equations

Constants of Proportionality

Mathematical relationships such as Hubble’s law are pretty common in life. To take a simple example, suppose your college or university hires you to call rich alumni and ask for donations. You are paid $2.50 for each call; the more calls you can squeeze in between studying astronomy and other courses, the more money you take home. We can set up a formula that connects p, your pay, and n, the number of calls

$$p=A\times n$$

where A is the alumni constant, with a value of $2.50. If you make 20 calls, you will earn $2.50 times 20, or $50.

Suppose your boss forgets to tell you what you will get paid for each call. You can calculate the alumni constant that governs your pay by keeping track of how many calls you make and noting your gross pay each week. If you make 100 calls the first week and are paid $250, you can deduce that the constant is $2.50 (in units of dollars per call). Hubble, of course, had no “boss” to tell him what his constant would be—he had to calculate its value from the measurements of distance and velocity.

Variables, constants, proportionality

Equations - used in physics to simulate behavior of nature - predictive

define variables and constants

Directly and inversely proportional

exponentials

example problems

Constants

Do not delete tables! Attribution tag!

Examples and Exercises

Further Exploration

• https://www.nist.gov/si-redefinition...urement-system
• https://www.nist.gov/pml/owm/metric-si
• https://www.nist.gov/pml/owm/metric-si
• To learn more about the relationship between mass, volume, and density, use this interactive simulator to explore the density of different materials.