Basic Graphs, Lines, and More ...
Graphing: why do we "graph"?
Graphing is a pictorial way of representing relationships between various quantities, parameters, or measurable variables in nature. A graph basically summarizes how one quantiy changes if another quantity that is related to it also changes.
Taking the perspective of a student learning astronomy or physics, graphs are useful because they can summarize a LOT of information into one picture. When scientists do research graphs are often the only way to represent data that has been collected. When one collects data one often does not know the exact relationship and interdependence between the various quantities that are being measured. A graph can give one an idea about how these variables change relative to one another. For this reason you will encounter many graphs in your astronomy courses and in textbooks.
For example, to illustrate how the temperature of the sun varies as one moves from the center to the surface and then off the surface, it is best to use a graph as follows:
This figure tells us that the highest temperature in the sun is observed to be at its center and that the temperature drops as we go towards the surface. The graph not only summarizes a lot of detail about how the temperature changes but it also gives us a nice mental image to "hang onto" in order to have some intuitive picture of a process.
The Basic Graph: Y vs. X
To start off with, we should look at some very basic graphs. The idea here is to understand what purpose a graph surves. This will make it easier for you to understand all the various plots that you will see presented in class and in your textbook. Generally, it's a good habit to plot things in science. It works best at explaining what one wants to get across.
Plotting is actually an extremely simple idea in its basic form. Here, and in most parts of the courses on basic astronomy, you will be looking at and analyzing plots of two variables at a time; this means that you'll have two quantities. Let us take an example. Suppose you have measured the apparent brightness of light from a variable star from day to day and have kept track of how the stars brightness changed over time (from hour to hour, day to day, or ...). Your two variable are the quantities that you are measuring, namely: 1. apparent brightness and 2. time. The idea behind graphing is to take your data and plot is as points on a set of axes. We usually call one variable the Y-axis and the other variable the X-axis. These axes are basically two lines that are prependicular to each other as shown here:
Notice that Y usually labels the vertical axis and X labels the horizontal axis, and the two axes are always plotted perpendicular to each other. So, in our example above, we can choose the brightness of the star to correspond to our Y-axis and the time variable to correspond to our X-axis. Plotting brightness vs. time gives us the following graph:
The blue curve in the figure reprsents the plot itself. That is, to come up with the blue curve you take each data point - which consists of a time and a brightness value - and you plot that point on the graph with the brightness corresponding to an amount along the Y-axis and the time corresponding to an amount along the X-axis. You plot all your points and if you have enough number of data points and they are closely spaced you can connect them to form a smooth curve, as the above example figure shows.
Simple Graphs: Lines, Periodic Functions, and More
The simplest kind of graphs you will encounter are those in which the relationship between two variables is linear. Linear relationship simply means that the the the values are related in a way such that if one variable is changed by a certain amount the other variable also changes by a constant proportional amount. We can symbolically write this as:
When variables that related in this way are plotted on an XY grid - as discussed in the previous section - the graph tuns out to be a straight line. The constant value that relates the changes and makes them proportional is the slope of the line in the graph. A typical linear relationship will look like:
Note that there can be different slopes associated with a line, so the above picture is only a shematic representation of a "typical case".
There are many situations in nature in which some quantity will change periodically with time; an example is the brightness of a variable star that was shown in the plots in the previous section. When we say that a quantity changes periodically with time we mean that there is a definite period in time that is associated with the observed changes; if you wait an amount of time then the quantity will have returned to its original value. Plots of periodic functions will typically resemble the following:
These types of periodic changes in values physically observable quantities are extremely common in astronomical processes and in our every day world. The earth going around the sun is an example of a periodic fucntion: it takes a period of about 365 days for the earth to complete a revolution and come back to the position it started at. The phases of the moon are periodic, with a period of about 29.1 days, as observed from here on earth. There are detail and complexities that come in when one looks at the real situations, but you need not worry about those at this point. You should understand what it means for a function to be periodic: namely, that the value described by that function repeats over a certain definite period of time.
Finally, you should also expect to encoutner exponential and power relationships (in the sense of algebraic powers). These relationships are also extremely common in nature. Again you will notice in such cases two physical variables whose mathematical relation can be written as:
The quantity a is a generic label for an algebraic power and the two variables are represented by x and y. The exact shape of the graph in such cases really depends on what the value of the power a is. This algebraic power a could have the value: 1.5, -3.2, 4, etc ... basically there are many possibilities! So, you should just remember that parameters and quantities we measure in the universe can be related this way and you should distinguish this from the linear and periodic cases discussed above.
There is a special type of power function which is very common in nature. There are many processes that show exponential changes. For instance, the intensity of light as it passes through a material of some density is seen to exponentially decrease as it passes throught the material. Species that are successful and have population growth show an exponentially increasing rate of rise in population. These types of changes are described by exponential fucntions an are written as:
y = ex
Here, y could be the population and x could be time, or y could be the intensity of light and x could be the distance it has traveled along a material. In general, exponential functions well look like the following:
In this figure, growing function, the power of the exponent is positive.
In the decaying changes the power of the exponent is negative.
Note that algebraically these kinds of functions are e raised to some power. What is this e? Well, it's a number and it turns out to be e=2.718281828 .... You need not worry about where this number comes from. It turns up everywhere and you will usually not be interested in the number or its origin, but the fact that it occurs and many exponential processes in nature have changes that are characterized by this base.