What is a Model in Science?
In this activity, you will be fitting different models to the
spectrum of a low-mass X-ray binary. But first you need to know what a
model is.
Scale model of the Sun and planets. In this model, the Sun is
10-inch diameter playground ball.
Models can take many different forms depending on what you want to
use the model for. For example, let's say you wanted to model the
relative sizes of the planets. The largest ball you have is a pilates
ball, which is about 65 cm in diameter. Using the pilates ball as a
model for the Sun, then you could pick up a pea to represent Mars, a
pinhead to represent Mercury, and a golfball to represent Saturn. These
would all comprise a model of the relative sizes of the planets.
However, the model doesn't encompass everything about the planets –
for example, it doesn't represent their relative densities or distances.
In this activity, you will be using a different sort of model. The
above example is a physical model – one that you can touch with
your hands. To model a low-mass X-ray binary spectrum, you will be using a
mathematical model. A mathematical model is an equation or set
of equations that simulate a physical process. You are probably
familiar with the equation for the force of gravity:
Fg = G m1
m2/r²
That equation represents a mathematical model of the force of
gravity.
How to build a simple mathematical model
Imagine that you notice that students that study more in your physics
class appear to get higher grades. Can you quantify this? Can you
determine how many hours you need to study to get a certain grade? For
example, does studying twice as much double your grade on an exam? Or
is the relationship between study time and grades more complex?
If you come up with a relationship that's close to reality, you might
be able to predict how many hours you need to study to get an A for the
semester.
To build a model, you need to start with some data. Let's say you
find out the following:
- Student 1: studied 1.5 hours, got 44% on exam
- Student 2: studied 3.5 hours, got 81% on exam
- Student 3: studied 5 hours, got 96% on exam
A plot of hours studied versus test score is shown below. As a first
guess, you might draw a line through the data, as illustrated below.
Graph of data set for test scores versus time studied. The line is an
initial guess at a model to fit the data.
The equation of the line drawn through the data can be found with
basic graphing techniques and can be written as:
y = (18.3) x + 12.5
where, y is the score (in percentage points) and x is the time spent
studying (in hours). This equation is your initial model
for the data.
With this initial model, we have assumed values for your two free
parameters: slope of the line and the y-intercept. The free
parameters tell your model how it can "wiggle" as you tweak the
model to fit your data. The best-fit values of those free parameters
help you interpret your data. In this case the free parameters are:
- The y-intercept, which tells you the grade you might expect to get
by not studying at all.
- The slope of the line, which tells you how many percentage points
you gain for each hour of studying. A shallow slope means that you have
to study many hours to gain just a few percentage points, while a steep
slope would mean that just a little study time will increase your test
score by a significant amount.
However, this model represents only your first guess at an equation
to fit the data. The next logical questions to ask are: "How good is
this model?" and "Can I make this model better?" Continue to the next
section, titled "How Good is the
Model?" to find out how to answer these questions.
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