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

1. 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.

2. 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

3. How far away (in km) is 10,000 light-years?

One light year is approximately 9.46 x 1015 m.
First convert the distance to meters: Something 10,000 (1 x 104) light-years away is
104 ly x (9.46 x 1015 m/ly) = 9.46 x 1019 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 1019 m / 1 x 103 m/km = 9.46 x 1016 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 106 km in diameter! This means that the scale of our model is 8.5 inches to 1.39 x 106 km.
1. How far in reality is something 5 inches away in our scale model?

8.5 inches represents 1.39 x 106 km, so 1 inch represents 1.39 x 106 / 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!

2. 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 109 years per month .
Invertebrate life appeared 600 million years = 600 x 106 = 6 x 108 = 0.6 x 109 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.