# Units Conversions and Scale Models

## Astronomy 10, Vista College - Spring, 2004

This handout gives a few examples of units conversions, which you
will have to do only rarely! But you should understand the reasoning
behind converting units (since this a useful everyday skill).
** Focus more on the first section**, since I will occasionally require you
to do some units conversions. The other sections are there
for completeness on the scale models that we constructed in class.
I won't ever ask you to do complicated scale model calculations.
## Units Conversions

### Simple Example

- How many inches long is a piece of wood 3.5 yards long?
To answer this, you need to know that there are 3 feet in a yard,
and 12 inches per foot. You're changing units from yards to inches.

So 1 yard = 1 yard x 3 feet/yard x 12 inches/foot = 36 inches.

Therefore 3.5 yards = 3.5 yards x 36 inches/yard = 126 inches.

- How many days is 36 hours?
You all know that there are 24 hours in a day. You're changing
units from hours to days.

So 36 hours = 36 hours / 24 hours/day = 1.5 days

- How far away (in km) is 10,000 light-years?
One light year is approximately 9.46 x 10^{15} m.

First convert the distance to meters: Something 10,000 (1 x 10^{4}) light-years away
is

10^{4} ly x (9.46 x 10^{15} m/ly) = 9.46 x 10^{19} m away.

Now convert from m to km. Since there are 1000 km in a meter (m),
then something 1000 ly away is

9.46 x 10^{19} m / 1 x 10^{3} m/km = 9.46 x 10^{16} km
away.

## Sample Scale Model Calculation:

A volleyball has diameter 8.5 inches. The volleyball represents the Sun
in this scale model of our Solar System. The Sun is really
1.39 x 10^{6} km in diameter! This means that the scale of
our model is 8.5 inches to 1.39 x 10^{6} km.
- How far in reality is something 5 inches away in our scale model?
8.5 inches represents 1.39 x 10^{6} km,
so 1 inch represents 1.39 x 10^{6} / 8.5, which is about 1 inch for
every 160,000 km. This turns out to be (since there are approximately 1.6 km
in every mile) about 1 inch for every 100,000 miles!

Something 5 inches away
in our model is really about 5 inches x 160,000 km/inch = 800,000 km away!

- How big is the Earth in this scale model? The Earth's diameter is
12,756 km.
Each inch represents (1,390,000/8.5) km (see above). How many inches does
it take to represent 12,756 km? 12,756 km is represented by

12,756 km / (1,390,000/8.5 km/inch) = 0.08 inches.

## Sample Cosmic Calendar:

Assume the Universe formed 12 billion years ago. Scale the history
of the universe to a single "year". Then the Big Bang happened just
after midnight January 1, and the present time is just before midnight
on Dec 31. Where on the calendar did invertebrate life appear? This
happened about 600 million years ago.

In this model of the calendar, each month represents about
12 billion/12 = 1 billion years.
1 billion = 1 x 10^{9} years per month .

Invertebrate life appeared 600 million years = 600 x 10^{6} =
6 x 10^{8} = 0.6 x 10^{9} years ago.

This means that on our scale calendar, invertebrate life appeared 0.6
months ago. An average month has 365 days/12 months = 30.4 days in it.

Which means that on our calendar invertebrate life appeared about 18
(~0.6 x 30.4) days ago.

18 days before Dec 31 is Dec 13.