Algebra Applications: The Hubble Telescope

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Narrator: The Hubble space telescope is our eye in the

sky, and it has delivered some stunning images.

The inner workings of the Hubble include a network of

mirrors and lenses.

A lens refracts light and focuses it on a point to form

an image.

There at three variables that are of importance with

any lens.

First, there is the focal length of the lens.

This is the point in space where a lens brings an image

into focus.

The focal length is usually described with the letter f.

The second variable is the object distance, in other

words, the distance between the lens and the object you

are looking at through the lens.

The object distance is usually described with the letter o.

The third variable is the image distance.

This is where the refracted light from the lens produces

an image of the object.

The image distance is usually described with letter i.

This equation describes the relationship among f, o,

and i.

For the Hubble, the focal length is a fixed quantity, so

we can rewrite the equation such that i is a function

of o.

This is a rational function that we can graph on the

TI-Nspire.

Turn on the TI-Nspire.

Create a new document.

You may want to save a previous document.

Create a graphs and geometry window.

At the function entry line input the function fifty-seven

point six x divided by the quantity x minus fifty-seven

point six.

Press ENTER.

You'll see part of the curve.

Press MENU and under WINDOW select ZOOM OUT.

Use the nav pad and move the pointer to the center of

the screen.

Press ENTER enough times so that your screen looks

like this.

Notice the horizontal and vertical asymptotes.

We can interpret these asymptotes in terms of the

abilities of the telescope.

For the function, the value of x equals fifty-seven point six

is not allowed since this would result in a zero in

the denominator.

As the value of x approaches fifty-seven point six from the

right the value of y approaches positive infinity.

As the value of x approaches fifty-seven point six from the

left, the value of y approaches negative infinity.

When you move an object in front of a lens the object

distance changes.

When the object distance equals the focal length this

corresponds to the asymptotic value.

At this value the lens does not produce an image.

This is why mathematically an object distance equal to the

focal length is not defined.

Because the Hubble space telescope was built for deep

space observation, then the object distances are vast.

You'll see that as x increases in value the y value, the

image distance, approaches the horizontal asymptote.

To see what the value is as x approaches infinity re-write

the equation in this form.

Now you can see that as x approaches infinity the term

with x in the denominator approaches zero, and the value

for the image distance approaches the focal length.

How well does the Hubble do with nearby objects, like the

moon and other objects in the solar system?

As you can see, as x approaches zero the image

distance increases or decreases in

value dramatically.

The image distance is not as stable as it is with

distant objects.

Here is another problem with nearby objects.

This diagram shows that the angle measure of an object

through a telescope is based on the size of the object and

the distance of the object, as well as the size of the

telescope lens.

The angle is measured in arc seconds where thirty six

hundred arc seconds equal one degree.

The Hubble's resolution is zero point zero five

arc seconds.

This is a very small angle measure and one that benefits

from long distances.

This is why galaxies are clearly visible through

the Hubble.

Galaxies are both huge in size and very distant, which take

advantage of the Hubble's optics.

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