Bad Astronomy and Bad Measures

Scale-You-Dont-need-a-stinking-scaleBy The Metric Maven

When I look back at my years of academic study, the academic institutions I attended seemed to view measures as something of secondary concern—or tertiary—at best. There is a strange irony that engineers and scientists can ascend academic educational levels from BS to PhD, and then become post-graduates, without ever attending a course contemplating or reflecting on how to best express numerical data. Data presentation is either thought to be beneath consideration, or simply knowledge acquired through osmosis.  Some of these academics then become popular science writers and presenters, without measurement expression introspection laying a glove on their consciousness.

Recently, Klystron provided a link to an article in Wired explaining why balloon castles go air-born when they encounter winds. The line from the article that jumped out was this:

We already have the speed to use: 25 mph, or about 37 feet per second. The density of air is 0.04 kilograms per cubic foot. (I’ll use kilograms for masses and pounds for forces and weights to avoid confusion.)

The author of this article, Brendon Cole, writes 0.04 kilograms instead of 40 grams and then mixes in Ye Olde English feet to create a single expression for density. Kilograms per cubic foot? WTF? Pigfish is introduced without reservation or thought. The author then decides to use Kilograms for mass and pounds for forces! The common pound is generally thought of as the avoirdupois pound, which is generally thought of as a mass. When referring to pound force, it is written as lbf. But 1 lbf = 32.17 ft*lbm/(s*s) where lbm is pound mass and lbf is again pound force. Clearly this is less confusing than simply using newtons?—really?

When I see such piss-poor mixed-up scientific expression in science writing I wonder if I’m the only one who notices. This person apparently gets financial compensation for this writing and cred as a “science writer.”

The density of air in Denver Colorado on a pleasant day is around 1000 grams per cubic meter, or about one Kilogram per cubic meter. Air density varies depending on temperature and pressure. Cole’s value when converted to metric is a very reasonable 1413 grams per cubic meter. Helium has a density of about 170 grams per cubic meter and, because of its much lower density, is clearly useful for filling balloons that will float upward. Cole chooses 0.04 Kilograms per cubic foot?—to make his calculations more accessible?

The fashionable term for today’s “science writers” is “science communicators” which I like, because compared with traditional science writers such as Isaac Asimov, L. Sprauge de Camp, Arthur C. Clarke and such, they don’t seem like they are serious about expressing science in a clear and cogent manner or prepared to do so. This is especially true when it comes to the presentation of numerical information.

Phil Plait is an astronomer and has also been involved in scientific skepticism during his career. As I’ve pointed out in the past, scientific skeptics can embrace a form of technical hubris when their use of the metric system or data presentation is questioned. Plait is the author of the book Bad Astronomy, and blogs at Slate. I find his topics and writing engaging and enjoyable, but his presentation of numerical information is at best ad hoc and devoid of introspection.

Chapter 3 of Plait’s 2002 book Bad Astronomy is titled “Idiom’s Delight: Bad Astronomy in Everyday Language.” It begins:

One of the reasons I loved astronomy when I was a kid was because of the big numbers involved. Even the nearest astronomical object, the Moon, was 400,000 kilometers away. I would cloister myself in my room with a pencil and paper, and painstakingly convert that number into all kinds of different units like feet, inches, centimeters, and millimeters. ….

The fun really was in the big numbers. Unfortunately, the numbers get too big too fast. Venus, the nearest planet to the Earth, never gets closer than 42 million kilometers from us. The sun is 150,000,000 (150 million) kilometers away on an average day, and Pluto is about 6,000,000,000 (6 billion) kilometers away. The nearest star that we know of, Proxima Centauri, is a whopping 40,000,000,000,000 (40 trillion) kilometers away! Try converting that to centimeters. You’ll need a lot of zeros.

Phil Plait does not seem to have any concern with using Olde English concatenated prefixing such as 40 trillion kilometers. I have written an essay that illustrates an elegant way large numbers in astronomy may be expressed and categorized using only metric prefixes. Solar System distances from the Sun to the planets are generally in the Gigameter range. The distance from the Earth to the moon is 400 Megameters according to Plait’s essay. The distance around the Earth is 40 Megameters, so one can immediately see the distance from the Earth to the Moon is about 10 times the distance around the Earth. The Earth is 150 000 Megameters from the Sun (or 150 Gigameters), Pluto is 6 000 000 Megameters from the Sun (or 6000 Gigameters). The nearest star, Proxima Centauri is in the Petameter range, or 6 000 000 Megameters distant. Why on Earth would anyone want to convert this value to centimeters!

He then offers a way out out of the centimeter difficulty:

There is a way around using such unwieldy numbers. Compare these two measurements: (1) I am 17,780,000,000 Angstroms tall. (2) I am 1.78 meters tall. Clearly (2) is a much better way to express my height. An Angstrom is a truly dinky unit: 100 million of them would fit across a single centimeter. Angstroms are used to measure the sizes of atoms and wavelengths of light, and they are too awkward to use for anything else.

American scientists always turn to centimeters as their go-to “scientific” pseudo-inch. Long time readers know I have a low opinion of centimeters. If you are a new reader, see, this, this and this or read Pat Naughtin’s epistle on centimeters or millimeters, or watch his video. The use of the Angstrom as a unit of measure places Plait squarely with non-SI usage, and on the side of unnecessary complication and confusion. If he kept with modern practice, he would have written his height as 1 778 000 000 nanometers versus 1.78 meters tall. Plait does not offer 1778 millimeters as a height, he rounds to the nearest centimeter without giving it a second thought. The pseudo-inch is very entrenched. His essay could have been about an optimum choice of metric prefixes and their simplicity, but Plait has not been prodded by his academic experience or elsewhere to even contemplate how he expresses numbers. It is examples like this that cause me to chuckle, and sometimes laugh out loud when I’m told that “US scientists use the metric system.” They use a strange mixture of cgs, mks and Ye Olde English.

Once the Bad Astronomer has established that angstroms are “too awkward to use for anything else” He carries on with his view of the situation:

The point is that you can make things easy on yourself if you change your unit to something appropriate for the distances involved. In astronomy there aren’t too many units that big! But there is one that’s pretty convenient. Light!……..

Plait then offers an enraptured colloquy that the Moon is 1.3 light-seconds from Earth, the Sun is 8 light-minutes from Earth and Proxima Centauri is 4.2 light-years. Then this:

The light-year is the standard yardstick for astronomers. The problem is that pesky word “year.” If you’re not familiar with the term, you might think it’s a time unit like an hour or day. Worse, since it’s an astronomical term, people think it’s a really long time, like it’s a lot of years. It isn’t. It’s a distance

The use of light-year by astronomers is simply confusing inside-speak. I’m going to defend the pesky word “year” in this situation. I have studied electromagnetism and its propagation for my entire multi-decade career. Light is an electromagnetic wave; it travels at 300 Megameters per second in a vacuum. The speed of light is the speed limit for sending information along wires and through interstellar space. Einstein was clear about this. The speed limit of the universe is 300 Mm/s or 1 080 000 Mm/h. The term light-year, light-hour or light-second make much more sense when thought of in terms of time. When we look at the farthest reaches of the Universe, we are looking back in time. The more powerful a telescope is, the further back in time it allows us to see. The Voyager spacecraft is about 19 light-hours from us. It takes 19 hours for a signal sent from Earth to be received by the probe. It then takes 19 hours to make the return trip. Light (electromagnetic waves) are the only way information is conveyed around the universe. Well, until gravity waves were detected, but they also travel at the speed of light. When, in 1604, Kepler saw the supernova which bears his name,  the stellar explosion which produced it had occurred about 20 000 years earlier. Expressing 20 000 light-years as a distance suppresses the more important interpretation in terms of time. Phil’s view:

I can picture some advertising executive meeting with his team, telling them that saying their product is “years more advanced than the competition” just doesn’t cut it. One member of the ad team timidly raises a hand and says, “How about if we say ‘light-years’ instead?”

It sounds good, I’ll admit. But it’s wrong. And more bad astronomy is born.

The subtitle of Phil Plait’s book Bad Astronomy is: “Misconceptions and Misuses Revealed, from Astrology to the Moon Landing `Hoax’.” Scientific skepticism involves questioning our most basic concepts, but scientists, like most people, often don’t. In this situation Phil Plait defends astronomical tradition over introspection. Could he for just one moment, think about the fact that perhaps the word “year” is equal to the word light in light-year, and time might be a better interpretation than distance. Isaac Asimov pointed out that as a unit the light-year is so small, that a sphere with a one light-year radius will not encompass our nearest star. Might not questioning or thinking about the assumptions and units astronomy uses to relate distance and time be an example of bad astronomy?—or simply ossified astronomy? The introduction of parsecs, light-years, AUs and such have an implied assumption for the reader that they are indispensable to Astronomers, and so they are units that the reader must learn to deal with despite an unfamiliarity with them. Scientific communicators and astronomers could use the metric system, which is the scientific standard of the world, but instead popular science writers embrace astronomical argot. Megameters, Gigameters, Petameters and such are too much of an intellectual imposition for delicate US readers, but parsecs, AUs and light-years are acceptable?

The chapter on measure ends with quantum dimensions:

In reality, a quantum leap is a teeny-tiny jump. The distances are fantastically small, measured in billionths of a centimeter or less.

So you might conclude that an ad bragging about a product being a quantum leap over other products is silly, since it means it’s ahead by only 0.00000000001 centimeters!

You might be surprised to find out that I have no problem with this phrase. I don’t think it’s bad at all! The actual distance jumped may be small, but only on our scale. To an electron it truly is a quantum leap, …..

I can only hope that when Phil Plait said that quantum leaps are “measured in billionths of a centimeter” he did not actually mean any scientific instrument would display such “units.” In my view this phrase is part of the problem. A billionth of a centimeter is 100 femtometers or 100 x 10-15 meters. This is five metric triads or 15 orders of magnitude smaller than a meter. He also reflexively chooses the pseudo-inch, aka the centimeter, as the base for his illustration.

An electron has a dimension on the order of 3 femtometers. Electrons shift between energy levels in atoms and not across exact locations. One can’t really say exactly “where an electron is” around a given nucleus. It is inside a sort of “probability cloud.” We can use a value called the maximum radial probability distance as a benchmark. It lets us “pretend” we have a solid location for the position of an electron. The classic Bohr radius for an electron in a hydrogen atom is about 53 000 femtometers from its single proton nucleus. When expressed this way, one can see that a 3 femtometer electron, which is 53 000 femtometers from its nucleus, is positioned at a very long distance in terms of its dimension. In the case of a hydrogen atom, the maximum probability electron radii for the first three energy levels are 53 000 fm (1S), 281 000 fm (2s) and 689 000 fm (3s). The distance from the 1s to 2s subshell is 165 000 femtometers. This is very large distance for an electron to traverse compared with its own 3 femtometer dimension.

The essay Plait wrote could have helped to educate the public and reinforce the use of the metric system for his professional audience. When I have asked why the metric system is not used in American science writing, I generally get an excuse that the author must write using units the general public understands. Strangely, this often involves centimeters. If the purpose of science writing is to qualitatively inform and entertain only, perhaps this would make sense, but to educate is part of informing an audience. Readers of popular science magazines are interested in the acquisition of new knowledge, why would the metric system be too large an imposition for this self-selected audience?

I meant for this essay to cull examples from numerous articles found in diverse publications, but the contents of Phil Plait’s Bad Astronomy chapter impelled me to follow it to its end and this essay expanded precipitously. The problem is not Phil Plait, it’s the entire pantheon of modern “science communicators.” They are a cohort of public intellectuals that praise numeracy and then dismiss the metric system. Phil Plait, Neil de Grasse Tyson, and Bill Nye are perhaps the best known examples. From a measurement expression standpoint, current science writing in the US is abysmal overall. Henri Petroski’s essay on paperweights in the July-August 2016 issue of American Scientist is a primer on how awful fractions are for expressing magnitudes. New Scientist, Scientific American, Discover, Astronomy and other popular magazines I read all are tone-deaf when it comes to using the metric system, yet claim to be performing a symphony of science. It’s time they tuned their instruments.

Related essays:

The “Best Possible Unit Bar None”

Long Distance Voyager

The Expanding Universe


The Metric Maven has published a new book titled The Dimensions of The Cosmos. It examines the basic quantities of the world from yocto to Yotta with a mixture of scientific anecdotes and may be purchased here.


The Expanding Universe

TelescopeBy The Metric Maven

After I learned it, I’ve made good use of the Whole Number rule in my technical work. But over the last eight years, I’ve found that now and then, I need to present data with a large dynamic range. What this means is that the numbers involved vary from a very small value to a very large value and span many metric prefix ranges. I’ve spent a lot of time trying to find a way to represent these values as intuitively as possible, and until recently all the options were, in my view, unsatisfactory.

Not long ago I was reading an essay by Isaac Asimov entitled The Figure of The Farthest. The essay discusses the  increase in the estimated size of the known universe from antiquity to 1973, when this essay was written.

Asimov starts with Hecataeus of Miletus (c.550 BC — c.476 BC) who wrote a world survey. The extent of the world he traveled was about 5000 miles across and thought to be a flat disk.


Hecataeus of Miletus Reconstructed World Map — Wikimedia Commons

Greek philosophers came to the realization around 350 BC that the Earth is probably a sphere. Eratosthenes of Cyrene (c.276 BC — c.195/194 BC) was the first to devise a method to measure the dimensions of the spherical Earth. He realized that incident sunlight at separate locations arrive at different angles. Using the distance between two points with a known measured angular difference, he estimated the Earth to have a diameter of 8000 miles.

The Greek astronomer Hipparchus of Nicaea (c.190 BC — c.120 BC) used trigonometric methods to compute the separation from the Earth to the Moon. His estimate was equal to 30 times the diameter of the Earth. Using Eratosthenes estimate for the Earth’s diameter, this distance is 480,000  240,000 miles.

Eighteen centuries passed before more refined astronomical estimates of distances were computed. The invention of the telescope in 1608 allowed astronomers to measure celestial values with much greater precision and accuracy. In 1671 French astronomer Giovanni Domenico Cassini (1625–1712) would be the  first to measure the parallax of Mars with reasonable accuracy. The model of the solar system provided by Johannes Kepler (1571–1630) was used to determine the distance to numerous astronomical bodies. The furthest distance measured was Saturn at 1,800,000,000 miles.

Edmund Halley (1656–1742) would compute the orbit of his eponymous comet to have a maximum distance from the Sun of 6,000,000,000 miles.

Measuring the distances to the stars proved much more difficult. A race was on to measure stellar parallax. A succession of star distances were finally measured. The distance to the star Vega was determined in 1840 by Friedrich Wilhelm von Struve (1783–1864). It proved to be the furthest at 54 light-years.

Countless stars remained without measurable parallax values.  Clearly the Universe was much larger than the distance to Vega. William Herschel (1738–1822) surveyed the number of stars in different directions and realized that it varied. He suggested they formed a flattened lens-shaped distribution that we now call our Galaxy.  It would not be until 1906 that Dutch astronomer Jacobus Cornelis Kapteyn (1851–1922) was able to use photographic techniques to estimate the dimensions of this stellar distribution. The largest dimension of the Galaxy was computed to be 55,000 light-years.

Harlow Shapley (1885–1972) would use a variable star called a Cepheid to determine the extent of our galaxy with increased accuracy. It was now thought to be about 100,000 light-years across.  Shapley further demonstrated that the Magellanic Clouds are outside of our Galaxy. The farthest extent of our Universe increased to 330,000 light-years by 1920.

The Andromeda Nebula became a source of scientific controversy. Was it inside of our Galaxy or outside?  In 1923 Edwin Powell Hubble (1889–1953) showed the Andromeda Nebula is indeed a Galaxy outside of our own, and the furthest extent of the known Universe expanded to 5,400,000 light-years.

Other fuzzy glowing patches could be seen, which suggested the Universe was much, much larger than the distance to Andromeda. By 1940, the maximum distance measured for the size of the Universe was around 400,000,000 light-years.

This value seemed to be a measurement limit and no further progress would be made. Then objects much brighter than galaxies were discovered. They were originally called quasars, but are now thought to be black holes which are swallowing nearby matter. The information they provided ballooned the maximum extent of the known Universe to 2,000,000,000 light-years. The year that Asimov wrote his essay, 1973, the farthest known quasar increased the size of the Universe to about 24,000,000,000 light-years.

In his essay, Asimov does not provide a table of these values. This is unusual, as Dr. Asimov had no reluctance to present numerous tables in his other essays and full length books. Each succeeding estimate of the Universe is separated with the explanations I’ve summarized above. I decided to create a table of Dr. Asimov’s data to use for illustration:

Table-1The values of the known Universe in miles become larger and larger and then Dr. Asimov shifts them to light-years. In my view this data has a perceptual discontinuity, and is not an acceptable way to present this data.

So what to try? Often in other tables distances are kept in provincial Kilometers, but clearly always starting with Kilometers would restrict an optimum starting value. I decided to try to categorize the data with a set of metric prefixes:

Table-2Clearly the situation is worse, the numerous sets of prefixes produce at least four perceptual discontinuities in place of the single one in Asimov’s data. I gave it another try where I attempted to minimize the number of prefixes and separate each set with a line skip:

Table-3Again, this is just a perceptual mess, with three rather than one discontinuity. I then thought about representing the lengths using a logarithm, and the knee-jerk way to do this is to use decibels. (Yes, I know that it is not usual to represent lengths as decibels, please humor me for a moment.) This yielded:
Table-4While this is continuous, it is also laughably decanted of all perceptual interpretation for most people. It simply seems to hide the magnitudes involved.

I started from scratch, and decided that it might be best if one chose a metric prefix which produces the smallest integer value possible in the data set. In this case it is eight Megameters. I then used only Megameters with standard three digit separations. This seems to be a useful way to present the large dynamic range data. One can see a large magnitude jump from 150 BC to 1671. The size of the universe was refined from 1671 to 1840 until another large magnitude jump occurred in 1906. The values increased without another large magnitude jump until 1940. From 1940 onward the increase was again without a quick jump in magnitude.

Table-5Still the table seemed to be missing something that might increase numerical clarity. I showed the above table to Sven for some brainstorming,  and he immediately had a suggestion. One could place the appropriate metric prefix at about a 30 degree angle above each of the three digit separations. I thought that using the metric prefix-base unit abbreviation might be best. Sven also thought that some light separation lines might be a good idea. My sense, from what I’ve learned from the book The Visual Display of Quantitative Information, was that this would distract from the data. When I implemented my thoughts, I ended up with:
Table-6My eye seems to be drawn to the metric prefixes at the top, which then act as a distracting interpretive boundary while I’m looking at the data. It struck me that a better alternative might be to  put the metric prefixes at the bottom:

Table-7While this is not perfect, it seems to help allow one to concentrate on the numbers with less distraction. My best suggestion for large dynamic range data is to:

  1. Use the smallest metric prefix that produces the smallest integer value possible for the smallest value in the data set.
  2. Tabulate the data with three order of magnitude separations spelling out the units at the right.
  3. Place the metric prefix-base unit abbreviations below each appropriate column.

I’ve pondered this problem for a long time. This is the first instance where a satisfactory form for large dynamic range data was obtained. This format may very well have been used before, but I don’t have an example (I’m sure my readers will let me know). I’m going to implement this format going forward, and continue to evaluate it. The use of spaces between the metric magnitude triads allows this format to work aesthetically. The column separation is immediately apparent. If commas are inserted as triad separators, the columns merge and become very difficult to cognitively distinguish. Independent of whether this is an optimum choice for large dynamic range data, it is simply not possible to create a table of this form using Ye Olde English units. It illustrates once again the superior nature of the modern version of the metric system’s units and methods.

Related essay:

Lies, Damned Lies and Scientific Notation


The Metric Maven has published a new book titled The Dimensions of The Cosmos. It examines the basic quantities of the world from yocto to Yotta with a mixture of scientific anecdotes and may be purchased here.