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### Measuring Distances in Space

```Date: 06/02/2008 at 09:53:28
Subject: How can we measure the distance from Earth to Mars accurate

How can we measure the distance from Earth to Mars accurately?  Space
is infinite--how can you measure something that is always moving?

```

```
Date: 06/05/2008 at 10:09:17
From: Doctor Ian
Subject: Re: How can we measure the distance from Earth to Mars accurate

It's a good question, and the answer is that we set up models for how
things move, and that allows us to use measurements to find things
like position and speed indirectly.

For example, suppose you're driving down the street, and a policeman a
block away uses his radar gun to determine that you're speeding.  How
can he do that, from a distance?

kind of light), and the frequency of the wave will change depending on
how fast you're going.  If he sends out one frequency and gets another
back, from the amount of change he can calculate your speed.  (Well,
HE probably can't, but the people who made the radar gun could, and
they included the necessary information in the device.)

Here's another example.  Suppose you know that a certain tower is 200
feet tall, and you'd like to know how far away it is.  You can measure
the angle that it makes above the ground (using a protractor, for
example), and then you can use trigonometry to relate the height, and
the angle, and the distance from you.  Since you know two of those,
you can get the third.  So you can tell how far away the tower is if
you already know the height; or you can tell how tall the tower is if

Can you see how we might be able to use this kind of reasoning to
figure out how far Mars is from Earth, even though it's far away and
moving quickly and we can't make any direct measurements?  (What if we
know about how big it is, and about what angle it subtends in the sky?
Then it's a lot like the tower example, isn't it?)

When you're dealing with bigger distances and speeds, the math gets a
little more complicated, and it requires a lot more imagination, but
the ideas are the same.  Take a look at this to see what I mean:

Size of the Universe
http://mathforum.org/library/drmath/view/56366.html

In the case of something like figuring out where Mars is, what happens
in practice is this:  We have a theory about what happens when two
bodies with known masses orbit each other.  So we can say:  If the sun
has such-and-such a mass, and the Earth has such-and-such a mass, then
the Earth should move around the sun on this curve, described by this
equation.  And we do the same thing for Mars.  Then we pick a time,
and we can calculate where Earth is, and where Mars is, and to get the
difference we subtract.

Now, how do we know that these equations are right?  Well, they're
never perfect, but we're always checking them using a process called
"least squares estimation".  That is, we may calculate that on some
particular day, at some particular time, Mars should appear in the sky
next to a particular star.  We go out and look, and it's close, but
not exactly where we thought it would be.  So we look for ways to
change the numbers in our model (how much mass everything has, the
sizes of the orbits, the locations of the observatories, and lots of
other stuff) and then keep running the changed model until we get the
smallest possible difference between what we predict and what we see.
Then we have a better model, and we can use that to get a new distance
from Earth to Mars... or from anything to anything else.

This is how we can do things like predict when an eclipse of the moon
will happen, even though it's years away; or determine the best time
to launch a rocket so it can get to Pluto with the least amount of
fuel; or predict when a comet is going to collide with Jupiter, so we
can make sure we get pictures of it.

It's also a great illustration of why scientists and engineers like
algebra and calculus so much.  Once you get some good equations that
describe something accurately, you can basically use it to predict the
future, whether that's where some planet is going to be at some
particular time, or how much weight it will take to make a bridge
collapse.  It's not QUITE as good as time travel, but it's probably
the closest thing we have to it.

Does this help?  If there's anything I said there that you don't
understand, let me know and I'd be happy to talk with you more about it.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 06/05/2008 at 11:47:01
Subject: Thank you (How can we measure the distance from Earth to Mars
accurate)

Wow, I never really expected an answer and that's a really good
answer.  I liked the least squares estimation explanation.  I never
heard about that before.  It sounds like a lot of work though to
figure out the answer.  I really appreciate you taking the time to
explain this to me.

```
Associated Topics:
High School Geometry
High School Practical Geometry
Middle School Geometry
Middle School Measurement

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