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Physics LibreTexts

1.6 Estimation

Many important calculations in science can be done by estimation. Estimation gives us a quick way to gain insight by using rough or approximate numbers. This is especially appropriate in astronomy, where many measurements have low precision and few important numbers are known to more than two or three significant figures. In estimation, we are satisfied with a precision of a factor of two or even a factor of ten, which is known as an order of magnitude. Estimation is perhaps the best way to come to grips with the enormous range of scales in the universe.

To take a frivolous example: how many pieces of paper would you have to pile up to reach the Moon? A ream of 500 pages of writing paper is roughly 40 millimeters thick. Although we might be able to make a more precise measurement of 39 millimeters or 39.6 millimeters, inestimation we make do with the round number of 40. The thickness of a single sheet is therefore

\[\dfrac{40}{500} = 0.080\; millimeters\]

The distance to the moon is 384,000 kilometers. Expressed in millimeters this is 384,000 times 1,000,000 or 3.84 x 1011 millimeters. So the number of pieces of paper required to reach the Moon is the distance divided by the thickness of a single page

\[\dfrac{3.84 \times 10^{11}}{ 0.080} = 4.8 \times 10^{12}\]

This number, nearly 5 trillion, is about the same order of magnitude as the sum of all the pages in all the world's books. Or imagine that the pieces of paper are instead dollar bills. The bank balance of most people would be a pile smaller than the height of a person. However, the worth of the richest person in the world would be a pile that reached several times around the Earth, and the United States budget would be a pile that reached nearly halfway to the Moon!

You could easily estimate how many times the Earth would fit inside the largest planet in the solar system, Jupiter. The Earth has a diameter of roughly 13,000 kilometers, and Jupiter has a diameter of roughly 140,000 kilometers. The volume of Jupiter is

\[ \dfrac{4}{3} π \left(\dfrac{D}{2}\right)^3 = \dfrac{4}{3} ( 3.14) (70,000)^3 = 1.4 \times 10^{15}\; km^3\]

If we imagine packing Jupiter with many versions of the Earth, such that they just touch each other like marbles in a jar, then each Earth will take up a space equal to a cube that just nestles around a sphere. The volume of each cube surrounding the Earth is D3 or (13,000)3 = 2.2 x 1012 km3. So the number of times the Earth will fit into Jupiter is given by

\[ \dfrac{1.4 \times 10^{15}}{ 2.2 \times 10^{12}} ≈ 600\]

Scientists use a wavy equals sign (≈) or another similar symbol (~) to mean "approximately equals" or "roughly equals." Jupiter dwarfs the Earth, and the largest storms on Jupiter are even bigger than the Earth.

One example will give us a sense of the vast distance between stars. The fastest man-made object is the Pioneer 11 spacecraft. This spacecraft has left the solar system several years ago and is traveling at about 110,000 kilometers per hour. How long will it take to reach the distance of the nearest stars? Alpha Centauri is 1.3 parsecs, or 3.9 x 1013 kilometers away. So the number of hours that Pioneer 11 will take to reach Alpha Centauri is

\[ \dfrac{3.9 \times 10^{13}}{110,000} = 3.5 \times 10^8\; hours\]

We can convert this to (3.5 x 108) / (24 x 365) ≈ 40,000 years. This is a sobering reflection on the capabilities of today's spacecraft. We will need new technologies before we can explore the stars.

Scientists try to only combine numbers with that have similar precision. Why? Because the result is governed by the number with the lowestprecision or the least number of significant figures. In other words, combining a great measurement with a lousy measurement will give you a lousy result. We can see this in our first example. Suppose that we know the distance to the Moon with a precision of 8 significant figures or an accuracy of about a centimeter. (In fact we do, using radar measurements!) Yet our measurement of the thickness of a piece of paper is only good enough to have 2 significant figures. This means that our estimate of the number of pieces of paper to reach the Moon should only be quoted to 2 significant figures — it is limited by our least precise measurement.

The way to get good at estimation is to practice. Try it! Just remember to express all the quantities you are combining in the same units. Mixing meters and kilometers or grams and kilograms is the easiest way to make a mistake that will throw your answer way off. If you are using a calculator, be careful to enter very large or very small numbers correctly in scientific notation. And if you want to estimate to only one or two significant figures, you don’t even need a calculator! This is what scientists mean by a "back of the envelope" calculation. Think of it like an artist doing a rough sketch or a color study before they undertake a detailed portrait. Scientists often take a staged approach, starting with an order of magnitude estimate then moving toward more precise calculations.