From left to right: Milton Humason, Edwin Hubble, Charles St. John, Albert Michelson, Albert Einstein, William W. Campbell, and Walter Adams during a 1931 visit by Einstein to Mount Wilson. Source: US Naval Academy, Nimitz Library

There is this interesting story about an ordinary back of an envelope. One day in 1931, Einstein (Albert, of course) went on a sabbatical leave from Princeton and stayed in the California Institute of Technology (Caltech). With his second wife, Elsa, he paid a visit to the Mount Wilson Observatory, where Edwin Hubble host the guest of honor. Both men are already famous for their respective discovery. Some 15 years before, in 1915, Einstein completed his theory of general relativity that changed our views on spacetime and made him a celebrity. Two years before, Hubble announced that his observations showed that galaxies are receding away from us at a speed proportional to their distance. This implies that the universe is expanding and might be an evidence for a moment of creation in the universe’s distant past. Hubble was already famous before, but this discovery increase his fame even more. At Mount Wilson, the Einsteins then were given a tour of the 100-inch (2.5 meters) Hooker telescope, then was the largest telescope in the world. The astronomers who worked there explains that this giant telescope was instrumental in recent discoveries about the nature of our universe. Elsa, looking not particularly impressed, replied, “Well, well, my husband does that on the back of an old envelope.”

Although his wife is not impressed with the 100-inch, Albert Einstein (left) was given the chance to observe with it, accompanied by Edwin Hubble (the one with the pipe) and Walter Adams. Source: Discovering the Quantum Universe.

Here is another one. During a symposium of astronomy, a theorist said, “You can collapse and make stars on the back of an envelope.” When I took the Star Formation course, in one of the sessions Ewine also said along the line of, “let us picture the condition by first doing some back-of-the-envelope calculations…”

What is it about this back-of-an-envelope calculations? What does the back of an envelope signify? Is it a symbol of mathematical prowess or does it symbolizes the ability to perform quick, simple, yet revealing calculations? From all the three stories we can temporarily conclude that back-of-an-envelope calculations involve theoretical physicists. Einstein might as well calculate the nature of the universe on the back of an envelope, and some short and simple calculations might be able to produce and collapse stars. A simple and rough calculations using previous assumptions and knowledge on the laws of physics might give an early sketch on the situation being tackled.

Stellar evolution is pretty much a solved problem, but star formation is still much of a mystery to astronomers. You might be able to collapse a star on the back of an envelope, but I doubt you can make a star within the space of an envelope. Photo credit: NASA, ESA, STScI, J. Hester and P. Scowen (Arizona State University)

In fact, back-of-an-envelope calculations are not meant to proof previous hypothesis. Although they are more than an educated guess, but they are far less than a proof. In theoretical fields, if you want to prove something, you need to perform more rigorous calculations. Back-of-an-envelope calculations, although useful to sketch down our early hypotheses and test our assumptions, are simply would not do. Einstein might did lots of back-of-an-envelope calculations to test his ideas and support his points, but to proved it he obviously had to perform a more rigorous proofing. Metaphorically speaking, if Elsa said that her husband explore the cosmos using a pencil and the back of an old envelope, I would rather say that he used the back-of-an-envelope calculations just to skim the beach in the cosmic oceans. To go further and explore the wider oceans, you need the strong vessel of rigorous calculations.

This might looked like a waste of taxpayers’ money, but they are useful in confirming astrophysical theories. Source: Mt. Wilson Observatory.

I would also have to stressed the importance of observations in astrophysics, and perhaps in any other fields of science. Theories would just be a dreamland fantasy if there were no observations that could confirm their predictions, and observations—no matter how good and meticulous they are—would just become trivial collections of fact if there are nothing that could explain their existence of those fact. Gigantic telescopes build on faraway mountaintops might seem like a waste of taxpayer’s money, but they are necessary to confirm Einstein’s jottings on the back of an envelope. However, Elsa’s derision of the role of observations are an old one. It has long been a public secret that sometimes there are conflicts between theorists and observers. Observers often rightly complain that theorists often treat observational literatures as if they do not exists and treat observers like unskilled labors. Is this the remnant of ancient Greek´s state of mind that explain things just by using their powers of deduction and logic without ever bother to observe? Using powers of deduction and logic are okay, by the way. But no observation whatsoever? Geez…

Hubble once said, “Observations always involve theory.” Source: PNAS.

Although the name suggest it, back-of-an-envelope calculations do not necessarily have to be performed on the back of a used envelope. The idea of back-of-an-envelope calculations are short calculations that try to picture or explain away a scientific problem in simple calculations that might fit the space of an envelope. Of course the space depends on which envelope size we are using :D, but those short and simple calculations could be performed anywhere or using a black/white board. Back-of-an-envelope calculations are not proof to any assumed hypothesis, however they are useful as an early sketch for the problem being tackled. It might not be useful in concluding any research project, but they are useful as a guide. An early probing using a back-of-an-envelope calculations might be helpful in deriving the expected order of magnitude. In fact, it might help somebody to become an efficient theoretical scientists.

In short, back-of-an-envelope calculations might not show your prowess in mathematics or physics, but it do show your ability to simplify complex calculations into several simple calculations that can estimate orders of magnitude. Symmetries could also be used to simplify the calculations even more. And so, back-of-an-envelope calculations might actually symbolize our ability to imagine the situation in your head and abstract the complex situation into simple building blocks. It can not stand on its own but it helps you answer the more complex problems. Furthermore, scientists believe that the universe as a whole actually behave in a simple and comprehensible way. Symmetry and beauty in any case is a subjective thing, but you know it when see it. Thus scientists prefer a simpler theory over a complex one. They might grinch upon seeing a complex theory and prefer the one that are simpler and more symmetrical. Back-of-an-envelope calculations usually worked out under these belief and that is why it is useful as a reality check. “You can collapse and make stars on the back of an envelope,” I never tried it myself, and although the remark sounds arrogant, but the theorist in the symposium might want to say, “please simplify the situation first. Science is not that complex and ugly.”

Written by Tri L. Astraatmadja

After living for 10 years in Europe as a Master's student, PhD researcher, and a postdoc, in 2016 Tri L. Astraatmadja moved on to the United States for a second postdoctoral appointment at the Department of Terrestrial Magnetism, Carnegie Institution for Science, Washington DC. He is now in his third postdoc at the Space Telescope Science Institute in Baltimore, MD.

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