Algebra Applications: Elliptical Paths

The Orion departs from the International Space Station

carrying its crew and the module that will land on the

surface of Mars.

But what is the path that the Orion will take to Mars?

It may surprise you that a direct path is not the best

way to get there.

To see why this is so let's look at the path of Earth and

Mars around the sun.

The path of each planet in orbit around the sun is

an ellipse.

An ellipse is a conic section that has two foci.

In terms of the path of the planets around the sun, the

position of the sun is at one of the foci.

An ellipse is defined as the locus of points such that the

sum of the distances from the two foci to any point on the

ellipse is a constant.

Take a string and tie each end to two points that will become

the foci of the ellipse.

Extend the string and use it to draw an ellipse.

This is the path that each planet takes, and it is the

path that the Orion will take on its trek to Mars.

On a coordinate grid, and ellipse has a width of 2B and

a length of 2A.

The values of A and B are used in the equation of

the ellipse.

Once again, note that an ellipse is a quadratic

relation, not a function.

An ellipse can have different values for a and b.

The eccentricity is a measure of how wide an ellipse is.

The equation for eccentricity also uses the value of

A and B.

This window shows slider values for a and b.

As the values of A and B change so does the value of

the eccentricity.

When a and b are equal the ellipse is a circle and its

eccentricity is zero.

As the values of a and b change the eccentricity is

greater than zero.

It approaches one but doesn't reach the value of one.

In fact, one way to understand conic sections is to look at

their eccentricities.

This table summarizes the values of e for the different

conic sections.

All circles have an eccentricity of zero.

An ellipse has an eccentricity in the range from zero to one.

A parabola has an eccentricity of one, and a hyperbola has an

eccentricity greater than one.

Since Earth and Mars have elliptical orbits, each

ellipse has an eccentricity in the range of zero to one.

This table shows the eccentricity for each orbit

and the distance from the sun.

Notice that the distance from the sun is measured in units

of earth orbits, so Earth's distance is one, and Mars,

which is farther from the sun than Earth, has a distance of

one point five two four Earth orbits.

We can use the data in the table to construct a graph of

each planets orbit.

Let's use the equation for eccentricity.

From the data table we know the values of e and a for

each planet.

We can solve for b to find the orbital equation for

each planet.

Let's start with the eccentricity for Earth.

Replace e and a with the values from the table.

Solve for B.

We get a value of B equals zero point nine nine nine

eight six Earth orbits.

This means that the equation of the ellipse for Earth is x

squared over one squared plus Y squared over zero point

nine nine eight six squared equals one.

Now lets find the orbital equation for Mars.

Replace e and a with the values from the table.

Solve for B.

We get a value of b equal to one point five one seven four.

This means that the equation of the ellipse for Mars is X

squared over one point five two four squared plus Y

squared over one point five one seven four squared

equals one.

Each quadratic relation can be written as two

quadratic functions.

This table summarizes the two functions.

The plus or minus sign indicates that one function

graph is above the x-axis and the other is below the x-axis.

Let's use the TI-Nspire to graph these functions and

analyze them.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a new graph window.

Input the first function for the orbit of Earth.

Press the DOWN ARROW to graph it.

This is the top half of the ellipse.

To graph the bottom half simply graph the negative

version of F1 of X.

Press the DOWN ARROW.

Now input the first function for the orbit of Mars.

Press the DOWN ARROW to graph it.

To graph the other half of the ellipse input the negative

version of F3 of X.

press ENTER.

Your graph may look like two concentric circles.

To get a more accurate view change the window settings by

pressing MENU, and under WINDOW ZOOM select

WINDOW SETTINGS.

Change the range of X and Y values to negative three and

positive three.

Click OK to accept the changes.

Now your orbits should look like ellipses.

The outer ellipse is the orbit of Mars, and the inner ellipse

is the orbit of Earth.

To clean up the screen press MENU to activate the

hide-show option.

Use the nave pad to click on each of the equations to

hide them.

The path that the Orion will take in traveling from Earth

to Mars is another ellipse known as a transfer ellipse.

In fact, in shifting from the Earth orbit to the transfer

ellipse the Orion will momentarily travel along

the hyperbola.

This diagram shows the transfer ellipse relative to

the orbits of Mars and Earth.

In fact, since you know the values of a and b for each

ellipse you can derive the equation of the

transfer ellipse.

The equation will be of this form, where h and k are the

coordinates for the center of the ellipse.

We leave this as a final exploration.

Use the TI-Nspire to find the equation of the

transfer ellipse.

As the Orion makes its final approach to Mars we can look

back and see that this voyage has really been a tour of the

universe of conic sections.

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