Chapter 1: A Sense of Scale

"10AM. 16 Peck Slip."

[1:] E-mail sometimes reminds me of the clipped cadences of British officers in old war movies. In this case, the message was a call to arms of sorts. It came from Dr. Neil Tyson, Director of the Hayden Planetarium, and informed me of the time and place of our appearance before the New York City Commission on Landmarks and Preservation. We wanted to knock down a protected building.

[2:] The massive granite facade of the American Museum of Natural History has presided over the Upper West Side of Manhattan for well over a century. The Museum, its echoing halls lined with an unimaginable menagerie of stuffed birds and animals, ancient Pacific war canoes, glistening minerals, and, of course, dinosaurs, has introduced generations of New York City school children to science. Also nestled behind its facade, on the north side of the building, was one of the Museum's more recent additions -- the sixty-year old Hayden Planetarium.

[3:] When the Planetarium was built in 1935, the New York night sky had long been ablaze with the illumination required by a city that never sleeps. The Milky Way was already a candy bar, not just a swath of opalescent light girding the night sky; at most a few dozen stars were visible on a clear night. Yet inside Hayden's dome ("the first example of sprayed concrete construction," the architectural historians solemnly informed us), the night sky came alive as the stuffed birds and the dinosaur skeletons never could for thousands of school children. The graceful sweep of the silver projector in the center of the room could carry them through the four seasons in a minute; to the southern hemisphere and back in less; or just upstate to an Adirondack hamlet where the stars still lit the evening sky. I have yet to meet a New Yorker between the ages of nine and ninety who hasn't taken these magical journeys, or weighed him or herself on Mars and Jupiter.

[4:] And we wanted to knock it down.

[5:] The inexhaustible supply of would-be politicians, self-proclaimed activists, and genuinely concerned citizens who live within 30 blocks of the Museum guaranteed that our quest would be an interesting one. The political connections and academic credentials of the Landmarks Commission members required us to prepare a compelling case. We felt we were ready.

[6:] The architectural firm of Polsheck and Associates had designed a breath-taking replacement. The Museum, under Ellen Futter's energetic leadership, had come up with the money to finance the $120 million project. We were going to keep the Martian scales and the Art Deco doors, although the historical sprayed concrete would have to go. In its place, we would erect a structure in which the full span of space and time would come alive -- a vantage point from which the visitor could gain an objective perspective on her place in the Universe.

[7:] Peck Slip. I had never heard of it, so I had to dig out my disintegrating street map of Manhattan from the bottom of the dresser drawer. Way, way downtown, in the tangle of streets laid out in lower Manhattan when it was New Amsterdam, I found it, projecting into the East River. A quay, really, no more than 200 meters long -- no need to leave early to find number 16.

[8:] Sixteen Peck Slip itself is a "landmarked" building. Once through the metal detectors (I guess some people get really passionate about old buildings) and up the stairs, I emerged into the hallway outside the Commission's imposing hearing room. Thirty foot ceilings, a table constructed at the expense of a small forest -- it was all on a grand scale, an appropriate venue for lofty civic principles and the sweep of history that gave context to the Commission's work.

[9:] Our presentation had been carefully prepared and thoroughly rehearsed. President Futter delivered articulate and forceful testimony about the Museum's vision of its role in the twenty-first century. The architects gave a sumptuously illustrated presentation of the new design, as well as a carefully reasoned explanation of why the current facility was hopelessly inadequate. Neil Tyson eloquently described both his own childhood experiences at the Hayden and his vision for its future with an ineluctable enthusiasm undiminished since his first visit. I was there to weigh in as Chair of the Astronomy Department at Columbia University with a dispassionate academic endorsement. Since academics feel most comfortable giving lectures, I delivered a lecture on the scale of space and time.

[10:] The centerpiece of the new planetarium's design was to be a smooth, 92-foot ball suspended inside a ten-story glass cube. The visual effect would be magical, but it was the possibilities for conveying a visceral sense of the scale of the universe that made the design irresistable. Imagine the nine-story sphere as the Sun. If constructed to scale, the Earth would be an 11-inch pale blue and green ball. We could hang it from the ceiling -- London and Dakar would be indistinguishable to the unaided eye. In the words of the great seventeenth-century Dutch physicist Christian Huygens, "A very fit consideration, and matter for reflection, for those Kings and Princes who sacrifice the lives of so many people only to flatter their ambition in being Masters of some pitiful corner of this small spot."

[11:] Now, the difference in diameter between the Earth and Sun is only a factor of 100; the Earth is roughly 8,000 miles across, while the Sun is about 800,000. The Earth's orbit is more than a thousand times the Sun's diameter. If we positioned the Earth to scale, then our 11-inch globe would end up orbiting the Planetarium Sun four miles away. I pointed out that this placed it just in front of the steps to City Hall, and suggested, following Huygens, that locating it there might serve as a gentle reminder to that building's occupants of their importance from the perspective of the Universe. Fortunately, the Commissioners (Mayoral appointees all) had a sense of humor.

[12:] In this New York-sized Solar System, Mars would be a six-inch pink ball in a school in Queens. Neptune, the most distant planet from the Sun, would be a glowing green sphere about 60 miles up the New York State Thruway with a subtitle: "You are now 2.7 billion miles from the Sun and 60 miles from the Hayden Planetarium." For the nearest star, we did have a problem. Another 90-foot sphere in central China would be sixty times too close.

[13:] And it was not just large scales the new Planetarium could illustrate. If the sphere represents an atom of carbon in your eyelash, that atom's nucleus is the size of a grain of sand. If the eyelash were made of atoms this large, it would be the width of Asia and stretch to the Moon.

[14:] The Commissioners got the point, signed the epitaph for the historical sprayed concrete, and cleared the way for a major new addition to the cultural highlights of New York City: The Fredrick Phineas and Sandra Priest Rose Center for Earth and Space. At night, the suspended sphere is indeed magical; by day, it serves as a window through which visitors can begin to grasp the scale of the Universe.

ARE HUMANS BIG OR SMALL??

[15:] Both, of course. In the last 100 years, science has developed a perspective on our place in the Universe by exploring particles both inside the atomic nucleus and billions of light years away in space. In doing so, we have uncovered the basic forces that govern the cosmos, woven them into the fabric of space and time itself, and reconstructed the history of the Universe and all it contains.

[16:] We begin, then, by exploring the range of physical scales that comprise the domain of science. This range is enormous, which leads to two immediate problems: 1) the numbers used to describe them are unimaginably large (and small), and 2) the distances lie far outside the realm of our experience and thus, by definition, don't comport with our "common sense." Albert Einstein's comment about common sense is relevant here: "[It] is that layer of prejudices laid down upon the mind prior to the age of eighteen." This chapter intends to help you cast aside those prejudices and look objectively at the full range of scales in space and time that shape our current understanding of the physical universe.

THE LIMITS OF PERCEPTION: The Scale of Space

[17:] First, let us examine the origin of our perceptions of small and large. What are the limits of the human senses? What is, for example, the smallest distance you can directly perceive?

[18:] The two senses one might think relevant to this problem are touch and sight. Let's start with touch.

[19:] Have a friend (and make sure it is a friend) do the following experiment. Get two very sharp pencils or a divider (the device used to determine distances on maps). Put a blindfold on and have your accomplice hold the points close together (less than a quarter of an inch apart), and touch you gently -- and absolutely simultaneously -- on your shoulder. Can you tell if it is one prick or two? Conduct a series of tests to determine the minimum distance at which you first feel two separate stimuli. Vary the spacing between the points until the prickee correctly identifies the two pricks as separate stimuli. To test the authenticity of the answers, the pricker should randomly alternate between one prick and two.

[20:] Even though the nerve endings being stimulated by the prick are very tiny (much too small to see with the naked eye), the elaborate process by which contact between the pencil's carbon atoms and your skin molecules is translated into a sense of being "touched" has a rather limited sensitivity. On your shoulder, where nerve endings are relatively scarce and the portion of the brain devoted to recording their activity is relatively small, distances less than roughly half an inch cannot be resolved. Other "more sensitive" areas of your skin might do better. Repeat the experiment (with the prickee blindfolded) on the index finger. What is the minimum perceptible distance here?

[21:] Take off the blindfold. Can your eye distinguish the two distant points that touch failed to record? If the answer is yes (and if it isn't you need glasses), perhaps the eye is a better tool for probing small distances. Let's test it.

[22:] Pluck a hair from your head. Stretch it between your fingers and hold it at arm's length. The width of the hair is still visible. Now have someone hold it up twice as far away. This is equivalent to looking at something half the width of a hair strand (an alternative approach would be to slice the hair in half lengthwise, but just doubling the distance is easier). Can you still see it? Double the distance again. Now, I suspect, your eyes can no longer discern it. This width, roughly one-quarter that of a strand of hair, is "too small" for you to see. [WHY?] Thus, it appears that the practical limit of human vision is a distance just a few times smaller than the width of a human hair -- a distance of about 0.1 mm [link to the System]. Does that make a hair the smallest thing that exists? For most of humanity's journey on Earth, the answer would be yes. It takes a philospher to speculate on the nature of things insensible to human vision. Or a scientist to develop techniques to measure much smaller scales.

[23:] What is the largest distance you can directly perceive? You can, of course, fly from New York to California, or even to Tokyo non-stop, and by the time you get off the plane, it feels like a very long way. But you have no direct perception of the distance. Indeed, sitting in a plane on the ground waiting for the de-icing machine for a few hours is nearly as uncomfortable, and you haven't gone anywhere. Trains and cars do not help much, even though the latter allows you more direct access to the instruments indicating how fast you are traveling. Reading an odometer conveys a sense of distance indirectly. For a direct sense of distance covered, only your legs will do.

[24:] What is the longest distance you can imagine walking? Take a week and hike the Appalachian trail (no, that is not one of the required assignments). If you are in good shape and don't sleep too much, you might make 100 kilometers (roughly the nearly unimaginable distance from New York to New Haven). That seems like a pretty good upper limit to the distance you can directly perceive.

[25:] Now, let's compare the small and large limits of your distance perception. If I wanted to cover the road from New York to New Haven with individual human hairs laid side by side across the road, how many would I need?

[26:] (1 hair width/ 0.1 mm) x (10³ mm / m) x 100 km x (10³ m / km) = 10⁹ hairs [HUH?]

[27:] One billion. That is, the ratio of the smallest distance you can directly perceive to the largest distance you might directly experience is a factor of a billion.

[28:] One billion is a large number. If you tried to count the billion hairs, starting tomorrow morning, and counted one, two, three, four, five... at the rate of twenty every five seconds, that would be 240 per minute, or about 100,000 if you did it 8 hours a day. You'd get to a billion (if you never took a weekend off) in 10,000 days or about 27 years. It takes a lot of hairs to cover the road to New Haven.

BEYOND DIRECT EXPERIENCE

[29:] Today we explore the natural world on scales far beyond what our senses can detect directly. What is a hair made of? If we peer down to lengths a billion times smaller than a hair's width, we are still on a scale too large to study the constituents of the atomic nuclei that anchor the hair's atoms. And if we expand our view to far beyond New Haven by a factor of a billion times greater than the distance to that city, we will not have reached much beyond the confines of our solar system. Another factor of a billion beyond that and we are still trolling around in the local Universe.

[30:] The full range of scales that science explores -- from the quark-gluon interaction inside a proton (or the first moments of the Universe) to the edge of the observable cosmos -- is about 100,000 billion billion billion billion or 100,000,000,000,000,000,000,000,000,000,000,000,000,000 .... which is why we will always use scientific notation in this course: the range is a factor of 10⁴¹.

[31:] Clearly this is not a number that is easy to visualize. That does not mean it is not a perfectly valid description of the realm of spatial scales we probe, but its size does mean that some analogies are needed to help us appreciate the scales of science. Here is an activity you can do on campus that will help (a least a little).

[32:] Get a tennis ball (chip in on a can with your classmates, scrounge around the courts in Riverside Park, whatever). This object is 6cm (0.06m) in diameter. To do this exercise, you will need a table of the diameters and distances of other things we will be discussing this semester.

[33:] Table 1: Diameters and distances of physical objects

[34:] 1. Using the tennis ball and readily available materials, construct a scale model of a Carbon atom using the tennis ball as the nucleus. Assume there are four electrons in the outermost (n=2) of two orbital paths around the nucleus and two electrons in the inner (n=1) orbit, and that the orbital radii scale as n². (The "carbon atom" link, above, will take you to a tutorial on basic chemistry.)

[35:] 2. How wide would a human hair be if made of atoms this size?

[36:] 3. How wide would the hair be if the whole atom were the size of a tennis ball? Could you see (with your eyes) the nucleus of such an atom?

[37:] 4. Using the tennis ball as the Sun (magically transform it by scaling it up by a factor of 10²³ or so from when it was a nucleus -- use abracadabra or whatever spell you wish), make a scale model solar system including the Sun, Earth, Moon, Jupiter, and Neptune.

[38:] 5. Where would you expect to find the nearest star if it lies to the west of the Sun?

[39:] 6. Shrink the solar system this time so Neptune's orbit just fits inside the tennis ball. Near what city would you find the Center of the Milky Way (assuming it lies to the west)? [U.S. map link]

[40:] While these models do not help a lot in visualizing numbers such as 10⁴¹, they should give you some sense of the scale of things that lie far beyond our direct perception. For example, real atoms do NOT look like the little solar systems in your high school textbooks -- the electrons orbit not a few inches, but a few blocks from the nucleus. Atoms are 99.9999999999% empty space. [WHY?] It is remarkable that you can't simply pass your hand through your desk, then, isn't it? On the larger scale, the solar system is a pretty isolated and tiny place compared to the Galaxy -- it makes it hard to believe that those tennis balls in Kansas, Anchorage, and Singapore (the very nearest stars) have much influence on us as we live out our lives on this little grain of sand 20 feet from our own tennis ball Sun.

[41:] Enough on the scale of space for now. Let us explore a scientific perspective on scales of time.

THE LIMITS OF PERCEPTION: Scales of time

[42:] We can begin again with the limitations our senses impose on the lengths of time we can directly perceive. The eye, for example, has a "flicker-fusion" threshold of roughly sixty flashes per second. That is, if a light is turned on and off around 60 times per second, one perceives it as a series of flashes, but if it is flipped on and off a hundred times a second it appears as a continuous, steady glow. One cannot, then, "see" anything happen on a timescale shorter than about 1/60th of a second. The ear has a similar limitation in distinguishing separate sounds -- at roughly 20 ticks per second the sound merges to a continuous hum. A fundamental limitation of all our senses is set by the speed at which signals travel through our nervous system (roughly 100 meters per second) and the rate at which individual neurons respond (a few milliseconds).

[43:] On long timescales, we have a very obvious limit -- our lifetimes. The amount of Bach your mother played to you in the womb notwithstanding, your direct perception of a "long" time is the number of years you have been alive. You might be able to extrapolate a bit -- to imagine what it must be like to have lived three or four times as long (as your professors have), but that's about it. You may have visited the California Redwoods and been told they were 2000 years old (or the Pyramids which are 4500 years old), but you certainly do not have a visceral feeling for this longevity.

[44:] Again, science has greatly expanded the timescales we can explore.

[45:] Actual experiments are more difficult (and tedious) in the temporal realm, so we will settle for some useful analogies. While your lifetime is certainly an upper limit to the length of time you can directly perceive, even that is a little hard to imagine -- you cannot, I suspect, remember much from your first five years, for example. A year seems a comfortable scale to deal with. You can probably recall where you were last year at this time, and you can reconstruct the major events in the intervening months. Thus, we will take a year -- one orbit of the Earth around the Sun, 365.25 days, 3.15 x 10⁷ seconds -- as our basic span of time. What are the temporal frontiers that science explores?

[46:] The upper limit on time for a cosmologist is the age of the Universe. That the Universe even has a finite age is a relatively new concept; from Galileo to Einstein, the prevailing view was that the Universe always had been, and always would be. But in 1964, a single observation overthrew that orthodoxy. In the intervening four decades, we have measured the Universe's age with increasing precision; we are now confident that we know the value to an accuracy of better than 2%: 13.7+/- 0.2 billion years (1.37+/-.02 x 10¹⁰ yr). We can reconceptualize the major events in the evolution of the Universe, the Earth, and humankind by creating a comfortable scale such that 1 year = 13.7 billion yrs. On this scale, 435 years pass each second.

[47:] The Universe, and time itself, began in the Big Bang. Let's think of it as the moment on New Year's Eve when the big ball hits the bottom of One Times Square. Within one one-thousandth of a microsecond of midnight, all the protons and neutrons that constitute the nuclei of all matter, and more than 80% of the helium in the Universe today, were produced. Fifteen minutes later (just in time for the next bottle of champagne), the first atoms were formed as itinerant electrons joined with free running protons and helium nuclei. The light from this epoch, 370,000 actual years after the beginning, is detectable today and provides some of the strongest evidence that we live in a Universe of finite age.

[48:] Around March 16, a large cloud of gas finally coalesced to the point where it collapsed into a spinning disk and started making stars; our Galaxy, the Milky Way, was born. Billions of stars blazed forth, then died. Some of these produced elements like carbon, oxygen, calcium and iron in their nuclear furnaces while alive. They then died in spectacular explosions that distributed these newly formed atoms throughout interstellar space where they became available to form the next generation of stars. Around 10:15 AM on September 1, a little clump of gas containing some of this freshly synthesized material began to collapse, and within an hour and a half or so, the Earth formed from the detritus left over from the Sun's birth.

[49:] Within a few weeks, complex molecules floating in the Earth's primordial ooze learned the trick of self-assembly and replication, and life emerged. But it was not until December 18 that a sudden (geologically speaking) blossoming of life forms occurred -- the Cambrian explosion. Flowering plants and complex animals appeared in the oceans and on the continents. On Christmas Eve (early December 24), land animals reached their pinnacle (in size at least) as the dinosaurs ruled a lush, tropical Earth. Then, just after 6:00 AM on December 27, disaster struck in the form of a ten-kilometer-wide asteroid that hit the Yucatan Pennisula in Mexico with the force of 30 million nuclear warheads, igniting wildfires worldwide, then plunging the Earth into a cold darkness that lasted for months and extinguished over half of all species of plants and animals worldwide. But over the next four days, this ecological disaster led to the emergence of new plants and animals, including some cute furry ancestors, the mammals.

[50:] And now it is New Year's Eve, December 31. At 10:15 PM, with the party in Times Square already several hours old, the first hominids appear on the plains of East Africa. Over the next hour and forty-five minutes, they evolve from upright apes fighting jackals for scraps from the carcasses of dead antelopes to, at 11:59:30PM, cave men emerging from a very long winter as the last Ice Age recedes on the warming Earth. The last 20 seconds of the year, it turns out, bring the stablest period of climate the Earth has seen in "days." Agriculture, then civilization, are born. The ball has already started to drop in Times Square.

[51:] At 11:59:49.4, with less than ten and a half seconds left in the year, those "ancient" pyramids were constructed for the burial of Egyptian pharoahs. Five seconds later, the rival schools of Aristotle and Democritus debated what the structure of matter is like on a scale too small to see. The next five seconds comprise the era studied in your other Core Curriculum courses. At 11:59:59.55, less than half a second to midnight, the longest enduring democracy of the modern world is proclaimed by the Declaration of Independence. And, with four hundredths of a second to go (less than the blink of an eye), you were born. Happy New Year, with best wishes for a new perspective on your sense of space and time.