GP 25 Web Book

Day 2
Lots of space

Bruno had little evidence that the Sun was actually a star, but he sensed that, if the Earth is a planet, stars are obvious candidates.  A century after Bruno, the vast universe was standard astronomy.

The sun borrows of the moon when Diomed keeps his word

Troilus and Cressida, Act V. Scene I

William Shakespeare (1564–1616)
The Oxford Shakespeare,  1914.

Figure 1: The rotation of the Earth causes stars to rise and set.  You can see the geometry. Viewed from the North Pole the Earth rotates counterclockwise.  A viewer (for simplicity in the drawing at the equator) sees the star rise when the eastern horizon is lined up with the star.  Six hours later the star is overhead at Z (at the Zenith or in general crossing the North-South meridian through the Zenith).  The star sets when it is lined up with the western horizon.  Astronomers plot the positions of a stars and planets on an imaginary celestial sphere surrounding the Earth (dashed circle).  The stars are far enough from the Earth that one sees half the sky at any one time.  (Lower right inset).

Primer on naked-eye astronomy.  Shakespeare, as well as his audience, was aware of the basics of naked-eye astronomy.  For example, the Moon shines by reflected light from the Sun.  We must understand the essence of medieval Earth-centered astronomy to understand the arguments that led to its demise.

Figure 2: The Ptolemaic system explained the observations available to the Greeks. Stars rise and set viewed from the fixed Earth.  The celestial sphere is a real object with the stars attached.  The stars rotate clockwise viewed from the North Pole.  They rise when they pass eastern horizon and set beneath the western horizon.  The ancients were aware that the Earth is small enough that one sees half the sky at any one time.

The Sun, Moon, and stars rise in the east and set in the west.  At the latitude of London, much of the sky never sets, but rotates visibly throughout the night around the pole star.  Copernicus, Bruno, and modern astronomy associate this motion with the rotation of the Earth about its axis (Figure 1).  In the Ptolemaic system, the celestial sphere of the Sun, Moon, planets, and fixed stars rotated en masse around a fixed Earth each day (Figure 2).  Modern astronomy retains the celestial sphere as a useful fiction for describing the positions of objects in the sky.

Figure 3: The planets rotate about the Sun counterclockwise viewed from the North Pole in the Copernican system.  I keep the diagram simple by showing only Venus, Earth, and Mars.  Venus revolves around the Sun faster than the Earth.  If the Earth is a point A in its orbit, special positions in Venus’ orbit are: IC, inferior conjunction between the Earth and the Sun; SC, superior conjunction, being on the opposite side of the Sun; and E1 and E2, maximum elongation, having the maximum angle from the Sun.  Transits occur at inferior conjunction.  The Earth overtakes Mars (between times T1 and T2) causing Mars’ motion to be retrograde against the background of distant stars.  The orbits are only mathematical entities, not physical tracks.  A spacecraft crossing the orbit of Mars sees nothing special.  Note that the disk size of Venus and Mars varies as they move nearer and further away from the Earth.

The Sun, Moon, and five naked-eye planets known in 1600 move with respect to the fixed stars.  This occurs since the Moon orbits the Earth each month and the planets and the Earth-Moon system orbit the Sun (Figure 3). The Sun and Moon both move steadily west with respect to the fixed stars.  Mercury and Venus orbit inside the Earth’s orbit.  Mercury, the symbol of the messenger of the Greek gods, darts back in forth from being a morning star and to an evening star.  It never gets far from the Sun and is not easy to see in the glare of twilight.  Venus also alternates between being an evening and a morning star.  It moves more slowly and gets far enough from the Sun to be the brightest object other than the Moon in the night sky. It casts a visible shadow in secluded locations on moonless nights.  It is visible in daylight if one knows where to look. Mars, Jupiter, and Saturn orbit outside of the Earth.  Their orbital periods (their years) are longer than the Earth year (about 365 and 1/4 days).  The outer planets generally move to the west in the sky. Each time that the Earth overtakes one of them (called opposition), the planet moves to the east in what is called retrograde motion.

The Ptolemaic system was adequate for describing the motion of planets in the sky, but became quite contrived when it represented retrograde motion (Figure 4).   In addition to revolving on its orbit around the Earth, each outer planet revolved on an epicycle that took it backwards each opposition.  Medieval opinion differed on whether the orbits and the epicycles were physical ”crystalline” objects or just convenient mathematical entities.

The procession of equinoxes was well resolved in antiquity. The Earth’s rotational axis processes like a slowly spinning top.  Its effects can be viewed in two equivalent ways.  The position of the pole of celestial sphere changes very slowly with time.  In a few thousand years Polaris will no longer be a good polestar. Equivalently, the position on the Zodiac of the Sun at equinox (when it is lined up with the Earth’s geographic equator) changes slowly with time, giving rise to the modern name of precession of equinoxes.  Ptolemy (ca. 85-165) attributed this to en mass movement of the stars in the sky.  Modern astrologers drone on whether signs should stay fixed with the stars of the Zodiac or rotate with the equinoxes. 

The reader of medieval literature encounters a fourth long-term motion, trepidation.  It was an artifact of errors in measuring procession.  The ancients did not resolve that the long-term motion is more complicated than just precession.  The shape and orientation of the Earth’s orbit changes slightly over time as does the tilt on the axis relative to the orientation of the orbit.

Figure 4: Mars rotates about the Earth counterclockwise viewed from the North Pole in the Ptolemaic system.  Retrograde motion occurs because Mars is on an epicycle that rotates clockwise.  Some carnival rides have seats that rotate in this way.  Many medieval astronomers considered the orbit and the epicycles to be real objects.  In practice a single epicycle did not adequately represent motion, the Ptolemaic system needed more epicycles and a dustbin of other contrivances.

The relative and absolute distances of objects in the sky were a key sticking point for the debate between Ptolemaics and Copernicans.  The Moon is obviously nearer to the Earth than the Sun, which it sometimes eclipses.  It also passes in front of the planets and the fixed stars, called occulting them.  It was well known to Aristotle (384-322 BC) that this indicated that the Moon was nearer to the Earth than the stars and the planets.  Transits of planets in front of the disk of the Sun are visible with the naked eye protected by smoked glass or safely with a pinhole camera, but only if one knows when to look.  Occultations of planets by the Sun are not visible to the naked eye.  (Astronomers reserve the term eclipse for when the disk angular diameters are comparable and use transit when the small disk passes in front of the large one and occultation when the small disk passes behind the large one.  They also use eclipse when a small object moves into the shadow of a larger one, as in a lunar eclipse.)  The ancients used guesswork to assign the slowly moving planets Mars, Jupiter, and Saturn to progressively further distances from the Earth outside the orbit of the Sun.  They differed on whether to put Mercury and Venus inside or outside the Sun or to have them circle the Sun.

Distance and Parallax.  The ancients used clever geometry [see Primer on Geometry] to determine the distances of the Sun and the Moon.  They had good estimates of the size of the Earth [see Do It Yourself Box on Navigation].  The methods involve parallax (Figure 5).  If you are not already familiar with the phenomenon, hold a finger in front of yours eyes.  Blink one eye and then the other.  Your finger will appear to jump back and forth on the background.  Now repeat the experiment with distant objects.  A distant foreground object will jump slightly on the more distance background, but your eyes cannot resolve the movement.   You need a larger baseline.  Parallax may become evident if you walk back and forth.

The Earth forms a useful baseline of thousands of kilometers.  One may observe objects simultaneously at different points on the Earth, geographic parallax.  Accurate timing to do this was impossible in 1600.  Alternatively, one can observe the relative motion of a nearby object like the Moon over a night with respect to the more distant stars (daily parallax).  The ancients used a mixture of these methods involving the eclipses of the Sun and the Moon.

Figure 5: Astronomers determine the distance to nearby stars by measuring the parallax angle agaianst the background of distant stars.  The baseline here is 2 AU the distance between the December and June positions of the Earth.  They record by convention the half angle P with a baseline of 1 AU.  The actual angles are tiny and the stars are moving with respect to the Sun.  It takes several years to get a reliable answer. Alternatively they could measure the change of angle relative to the Earth's surface. A star that passes directly overhead at the zenith in December will be slightly off the zenith in June. This method fails because of the aberation of light. Starlight appears to come (slightly) from the direction the Earth is traveling, like rain striking the windshield of a moving truck.

The disk of the Moon is a half degree of arc across, usually just big enough to totally eclipse the Sun, also about a half degree (Figure 6). The Moon casts an inner shadow, called the umbra where it totally covers (eclipses) the disk of the Sun and an outer shadow, the penumbra where part of the Sun’s disk is visible (called a partial eclipse).  You can see the effect yourself by projecting a flashlight on an object narrower than its beam or by totally or partly covering a distant object with your outstretched hand and closing one eye.

Figure 6: The Moon casts an inner shadow of total eclipse, the Umbra and an outer shadow of partial eclipse the Penumbra (left).  The actual disks of the Sun and Moon are about a half degree.  The Moon and the Earth are at the true relative scale and distance (right). At totality the umbra is a tiny spot on the Earth.  The Sun must be somewhere in the cone formed by the umbra and the disk of the Moon.  If the Sun is placed too close to the Earth in the diagram, the entire Earth ends up in the Penumbra, which does not happen.  This is evidence that the Sun is in fact very far away.  The line A and C are the essentially parallal as are the lines B and D.

A total or near total eclipse cannot be missed nor quickly forgotten.  In fact, it panics the ignorant.  Totality is brief and the band of totality narrow.  This information was quickly gathered in antiquity.  The band of the partial ellipse is relatively narrow (Figure 7).  The ancients sometimes gathered this information in the time of peace.  In contrast, a partial eclipse would cover the whole side of the Earth if the Sun were nearby (Figure 6).   As the Sun is actually much further away than the Moon, the Moon’s radius is given by the projection of a line from the edge of its umbra to the opposite edge of the penumbra on the disk of the Earth (Figure 7).

Figure 7: The eclipse of the Sun with the Earth and Moon and their separation are drawn to scale.  The Sun is very far away.  The diameter of the Moon is the distance D between the edge of the Penumbra and the opposite edge umbra as projected on the disk of the Earth.  It was hard to precisely determine this distance in antiquity but they got about the right result.

The Earth’s umbra completely covers the Moon during a total lunar eclipse indicating that the Earth is the larger body (Figure 8).  (An astronaut on the Moon then would see a total eclipse of the Sun by the Earth.)  The time for the Moon to pass through the umbra and the penumbra and the curvature of the umbra and the penumbra give the relative sizes of the Moon and the Earth.  The ratio of the umbra diameter to the penumbra diameter constrains the relative distances of the Sun and the Moon.  In practice, all the ancients could tell is that the Sun is very far away.  The line between the edges of the umbra and the penumbra then gives the Earth’s diameter.

Figure 8: The eclipse of the Moon with the Earth and Moon and their separation distance drawn to scale.  The Sun is very far away.  The diameter of the Earth is the distance D between the edge of the penumbra and the opposite edge of the umbra.  It was easier to use the curvature of the umbra on the Moon in antiquity as it is at present. With some geometry, the Earth's diameter is the umbra diameter plus the Moon's diameter because the disk diameter of the Sun and the Moon are both half a degree. Such methods give the diameter of the Moon as a fraction of the diameter of the Earth.  The ancients got reasonable answers.

Once one has the diameter of the Moon as a fraction of the Earth’s diameter, it is a simple matter to use the fact that the disk diameter of the Moon is a half degree to get the Earth-Moon distance.  Greek estimates were between 20 and 30 (the modern value) Earth diameters.  They knew that the distant Sun is much bigger than the Earth.

The size and distance of stars.  The Earth is too small to make a useful naked-eye baseline for objects further away than the Moon.  Greek and medieval astronomers realized that the Sun is many Earth diameters away and that the stars should show parallax if the Earth in fact orbited the Sun (Figure 5).  This parallax would occur even if all the stars were the same distance away on a crystalline sphere.   Then a star that was vertical (called the zenith by astronomers) when it passed a north-south line in the sky in June would no longer be vertical in December.  No annual parallax was ever detected.

Bruno and Galileo were well aware of this observation but they (like the Greek astronomer Aristarchus) contended that the stars were so far away that one couldn’t see parallax.  They had trouble making a convincing argument beyond that their system was simpler and that it seemed silly for the larger luminous Sun to circle the smaller Earth.  To proceed, one can make the trial hypothesis that the Sun is a typical star. One can independently measure the disk diameter and the luminosity of a star and compare these to see if the luminosity per area is comparable that of the Sun.  The ratio of the disk diameter of the Sun to the disk diameter of the star is then the ratio of the distance of the star to the distance to the Sun.  With a little geometry this gives the parallax angle.  The assumptions are not quite true. Some of the brightest stars in the sky are much more luminous and much wider than the Sun.  There are also many dimmer and smaller stars, some nearby, but no one in 1600 would start by looking at them.  It turns out the nearest star system, Alpha Centauri, contains two sunlike stars and one small faint star.  As we see below, one does get tolerable estimates, all with the naked eye.

Little got done immediately after 1600.  The invention of the telescope distracted astronomers.  Worse yet, the ruckus awakened the Church to the dangers posed by the Copernican system.

A new view of the sky.  Venice, May 1609.  A university professor hears a reliable account of a new device that lets one view distant objects.  The spyglasses are already on sale in Paris, but Galileo (1564-1642) is impatient.  He buys spectacle lenses from a local optician on his return to his home in Padua and makes some for himself.  The three-power spyglass is inadequate.  By trial and error, he determines the relationship of the magnification to the focal lengths of the lenses.  The opticians are a hidebound medieval guild.  They cannot supply stronger lenses.   Galileo figures out how to grind better lenses on his own.

His first success is an 8-power instrument, comparable to modern binoculars.  He must act quickly before it is a lira a dozen.  Always a semi-successful self-promoter and always short of cash, he presents his instrument to the Doge of Venice. He asks that his salary be increased.  This happens, but he is told that he will get no further raises.

By November, he has a 20-power instrument.  Fatefully he points it at the sky.  He sees shadows cast by relief on the surface of the Moon and images craters.  He sees that vast numbers of dim stars become visible.  Then he points the instrument at Jupiter. He resolves that it is a circular disk and hence a spherical body.  A line of 3 moderately bright “stars” appears near the planet.  He records them, but does not give it much thought.  The next night the stars are on the other side of the planet.  This is not expected since Jupiter’s motion is retrograde.  His interest is piqued.  Over several nights he finds that there are in fact four objects that circle Jupiter.  That is, there is a separate center of rotation independent of the Sun and the Earth.  The new objects are the moons of the planet Jupiter.

Speed is of the essence.  Others will soon have nice telescopes.  Galileo rushes a book, Sidereus Nuncius (The Starry Message or Messenger) with such haste that he does not get the distance from the Earth to the Moon right.  Simultaneously obtains a new patron by naming his new moons after the powerful Medici family in Florence and a no-duties professorial position to boot.  The Medici make him an international hero.  They send good telescopes and his book to the patrons of science, the crowned heads of Europe.

Galileo is quick to voice support for the Copernican system.  If Jupiter and its four moons orbit the Sun, so can the Earth and its moon.  Other discoveries quickly follow.  Galileo finds that Venus has phases like the Moon, which is expected from the Copernican system.  He finds that the disk size of Mars varies as expected (Figure 3).  All is there for all to see.  (You can easily see numerous stars with good binoculars and the moons of Jupiter if your hands are steady.  A small telescope nicely brings in the phases of Venus and shows the planets as disks.)

Trouble from a safe place.  Prague, 1610.  The Imperial court astronomer is troubled by news from Italy.  An Italian astronomer has found four new planets.  If they orbit another star. Bruno was right.  The Sun is one of an infinity of stars.  His friend Johannes Wackher forcefully points this out.

Finally the news arrives in the form of Galileo’s book.  Kepler is relieved and sends back a generally positive critique to Galileo.  He is not bothered by an infinity of stars, but he is sure that the Sun is not one of them.  He sets out to disprove Bruno’s “dreadful” theory once and for all.  His reasoning is simple: the disk diameter of stars is about one minute of arc, a 30th of the disk of the Sun.  This has been known (incorrectly) from antiquity. It is obvious (correctly) that the Sun is much brighter than 900 bright stars.  (The simplest way to see this is with a pinhole (Figure 9).  The sunlight is dimmed by a factor of 900 when the diameter of the spot is 30 times the diameter of the pinhole.  The spot is far brighter than starlight.)  Like modern science, Kepler’s argument stems directly from the observations.

Besieged by questions from all quarters, Kepler publishes his letter as a book, Conversation with the Starry Messenger.  The cat is out of the bag.  Soon everyone with even a casual interest in the sky is aware of Bruno’s infinity of worlds.  Kepler’s argument is weak because Galileo is already showing that the apparent disk diameter of stars narrows with each improvement in the telescope.  Only the planets show resolvable disks.

Rather than killing Bruno’s hypothesis, Kepler links it forever with the Copernican system.  There is no way the Church can suppress the telescope.  They do not even think of trying. It is too useful for military and navigational purposes.  It is a fine toy for the trendy.  There is no way to keep it pointed away from the sky.  The moons of Jupiter may have practical applications.  Galileo computes accurate timetables.  The occultations and transits of the moons occur at essentially the same time everywhere on the Earth.  They are a potentially useful clock for finding longitude (See [see Do It Yourself Box on Navigation), an important task in the age of exploration.

Figure 9: A pinhole camera produces a spot of dimmed sunlight.  The spot is as bright as sunlight would be viewed a distance in AU equal to the diameter of the spot divided by the diameter of the pinhole.  The pinhole camera produces an image of the Sun.  It is a safe way to view a solar eclipse.

The Church must react carefully. Its cozy universe rests on the fixed Earth. The Church defends at that point, not at the roomier Sun-centered universe of Copernicus and Kelper.  Explicitly condemning an infinity of worlds or that the Sun is a star will serve only to make even more people aware of these simply expressed ideas. (This is a little like the folly of publishing an exquisitely detailed antipornography law.) Galileo has not advocated the former or even mentioned the latter.  He has not cited Bruno.  He is a devout practicing Catholic.  The Churchmen set a trap to move the game to their home court with home referees.  They condemn the Copernican system on biblical grounds, like God could not have stopped the Sun for Joshua if the Earth moved.  These untestable assertions are better suited to the Churchmen than ignorantly going after Galileo’s physics and astronomy. Galileo does not let their attacks pass.  He responds in widely circulated private letters that they are vestiges from a simpler time and that the Bible should not be taken literally, but sticks to generalities.  He fears that the Church is about to impose doctrine that will be eventually (as it turned out within that century and conclusively by 1725) disproven by physical measurements.  His friend, the Carmelite priest Paolo Antonio Foscarini (ca. 1565-1616), publishes a detailed account of why Copernicanism does not conflict with the Scriptures. (Like Kepler, he contends that the Sun is much more luminous than a star.)  They are treading with their full weights on dangerous ground.  Bruno had unsuccessfully raised this dichotomy between religion and philosophy as a defense, but Galileo is confident that he can convince authorities in Rome.

Forever amber.   1616. The Cardinals and the Pope treat Galileo politely.  He is an international hero.  As a layman, he enjoys more leeway than Bruno and Foscarini.  Church dogma is intricate.  It is easy to fall into heresy. As a practical matter, the faithful must be allowed to recant and repent. In Italy, the dungeon, the stake, and the rack are reserved for the recalcitrant. They forbid Galileo from holding or defending Copernican doctrines, but do not specifically mention him in their public edict.  It is still OK to use the Copernican system as a useful fiction in calculations.  Whether they explicitly banned him from teaching the Copernican system will become an issue at his trial. Galileo is free to continue with astronomy as long as he does not advocate that the Earth circles the Sun.

But it is too late for the Church to drown Copernicanism.  By now, the populace is familiar enough with it that Foscarini needed only a cursory description in his book. Like ordering back the tide, the Church bans Copernicus’ book until it is fixed.  This has little direct effect on the street because only the most mathematical can understand it anyway. The Church allows it to be printed with a few changes four years later. (They totally ban it in 1664.) They ban Foscarini’s book outright and he dies soon after. This temporarily silences Italian proponents of Copernicanism.

1629. Galileo has managed to outflank the restrictions imposed by the Church.  He has spent endless sessions with the censors preparing his Dialogue Concerning the Chief World Systems.  He cleverly selects the names of his characters. Salviati and Sagredo are safely dead colleagues of Galileo. Simplicio (490-560, Simplicius in Latin) is an Aristotlean philosopher, safely well known to the censors. Simple can mean compact, elegant, and lucid, like with Newton's law of gravity. But it can mean simple minded. (Years later this view of Aristotleans will become universal; our word dunce comes from the Renaissance belittling of the followers of the Aristotlean John Duns Scotus (ca., 1265-1308).) Galileo cannot advocate the Copernican system, but there is nothing to prevent him from making his advocate of the Ptolemaic system, Simplicio, from living up to the second meaning of his name.  He is far more knowledgeable than the censors and is a master at incomplete arguments.  He leaves out the punch line at a point that fools the censors but lets the curious fill in the blanks.

His arguments about the disk size of stars are a tacit reply to Kepler’s Conversation.  He notes that the by-now-common telescopes show the planets as disks, but do not resolve the stars as disks.  He notes than the brightness per disk area of Venus, which is near the Sun, is much greater than the brightness per area of Jupiter, which is distant from the Sun.  Mathematically, the Sun can be considered to illuminate an imaginary sphere at some distance R from the Sun of an object.  The light per area on the sphere is thus proportional to the inverse of that radius squared.  The reflected light from the object (if it is much more distant than the Earth from the Sun) illuminates another imaginary hemisphere (containing the Earth) about radius R.  The light per area on the hemisphere is also proportional to the inverse of that radius squared. The net effect is that the reflected light from a distant object decreases with the inverse distance to the fourth power.  (The German battleship Bismarck was sunk because its captain was ignorant that these facts apply to radar. The Bismarck was still detecting British radar and did not keep radio silence because the captain assumed they were being watched anyway.   The ability of the Bismarck to detect British radar decreased with distance squared, but the ability of the British to detect the Bismarck decreased with distance to the fourth power.) The light from an object of constant disk angle decreases inversely with the radius squared.  Equivalently, to provide constant brightness as observed on the Earth, the disk angle must increase with radius squared.

Galileo points out that the brighter stars are much further away than Jupiter but about as bright.  For the stars to shine by reflected light from the Sun, their disks would have to be huge instead they are too narrow to resolve even with a telescope.   He then points out that a telescope is unnecessary. One can constrain the disk sizes of stars with the naked eye.

Galileo tries using a rope to do this [see Do It Yourself Box on Measuring the Disk Angle of Vega]. He walks away until he can see the star on both sides of the rope and computes angles accounting for the finite width of the pupil in his eye.  He gets 5 arc second for the bright star Vega.  (This result seems reasonable for an actual experiment.  If he used a 1-centimeter diameter rope, with a little calculation, the focal point needs to be 400 meters from the rope and the observer about half as far.  You can do it yourself with a planet.)  Galileo does not dare to push the argument any further, leaving the question of whether the Sun is a star unmentioned, but within the grasp of alert readers.

Despite the approval of the censors, Galileo has gone too far for the Church.  His just estranged friend, the Pope, feels betrayed by the book and humiliated by Simplicio’s final speech that man cannot fully comprehend God’s infinite power.  It is obvious to all that Galileo’s book advocates Copernicanism in Italian, the language of the streets.  Galileo is called before the inquisition in Rome.  He knows from Bruno’s fate that these people are serious.  Practical men hold the day and reach a plea bargain.  The Church avoids having to burn alive an internationally known scientist for all of Europe to see.  Galileo knows he has mortally wounded geocentric astronomy and that the Church will not be able to put his message back in the bottle.  By the time his book is banned, there are few unsold copies to seize. Galileo’s life ends in house arrest where he founds mechanics and material science.

The chill spreads far from Galileo’s villa.  René Descartes (1596-1650), a firm Copernican who is sure the Sun is a star, defers publishing his magnum opus on The World. He eventually seeks intellectual safety in Sweden.  Like Copernicus before him, he perishes and then publishes.  (Implicit is that they did not fear the powers of Churchmen in the hereafter but did fear them in the here and now.) The Dialogue remains banned until 1835. Galileo is for the most part rehabilitated in the twentieth century.

Retrospective and lunar occultations.   Modern astronomers have studied Vega.  It is 1.6 million times as far away from as the Sun or 25 light years.  (From now on, I use the astronomical convention of calling the distance between the Earth and the Sun, an astronomical unit, abbreviated as AU.) Vega’s diameter is about twice that of the Sun.  Its disk angle is thus the Sun’s disk angle one half degree or 1800 arc seconds times 2 divided by the distance ratio 1,600,000.  This yields 0.002 arc sec so that Galileo was off by a factor of 2500.  His 1-cm diameter rope would have to have been 1000 km away to work.  

Galileo method failed because he did not understand that bending (refraction) of light in the atmosphere causes stars to twinkle. That is, subtle variations of temperature in the atmosphere act like moving lenses. The effect is like desert mirages, only weaker; it is similar to a point at the bottom of a wavy swimming pool appearing to move around.  The path that the light takes through the air moves around (by less than a meter) too fast for the eye to resolve. The point of light comes in from slightly different directions at each “instant”. The disks of planets are larger than the range of incoming directions so they do not twinkle.  Viewed with modern photo-electronics, the instant image of a star is effectively a point. Modern telescopes combine the point of a “guide” star to align numerous brief images, using adaptive optics to continually adjust the mirrors.  If no guide star is in the view field, they may use lasers to create an artificial point source 90 km above the surface.

Galileo could have done much better using lunar occultations than he did with his rope. (There is no documentation on this matter so he most likely did not think of it. Had he thought of it, he could have easily passed the matter off as highly technical astronomy and left the inferences to the reader. If the church had tried to suppress the idea, it would now be a cause célèbre.) It was well known by the time of the ancient Egyptians that stars disappear behind the disk of the Moon.  About an hour later, the star emerges on the other side of the Moon (unharmed to the relief of the ignorant).  Watching an occultation sparked Copernicus’ interest in astronomy.

It is easy to constrain disk diameter with occultations (Figure 10).  The Moon moves across the sky with a known rate.  It circles the sky, 360 degrees each with 60 arc minutes and each with 60 arc seconds each month (crudely 30 days times 24 hours with 60 time minutes and 60 time seconds).   That is, (360*60*60)/(30*24*60*60) or 0.5 arc second per time second.  A fixed star blinks off in our reaction time of about 0.1 time second so its disk diameter is less than 0.05 arc second.  Good timing is not essential to get this quick estimate and the gist of its implications.  The disk diameter of the Sun is 1800 arc second.  If we assume that our star has the same diameter as the Sun, it is at least 1800 divided by 0.05 times as far away, or 36000 AU.  The parallax angle is smaller than 1/36000 radians [small angle formula in Primer on Geometry] or 6 arc seconds which is way too small to be resolved in 1629.  This suffices to silence lack of annual parallax as an objection to the Copernican system. 

Figure 10: You can watch the occultation of a star by the Moon.  The star blinks off quickly as it is covered by the limb of the Moon.  This allows one to constrain the disk diameter.  The star dims more slowly when its appears to approach the Moon at a near-grazing angle below.  The dimming time then can be estimated with the naked eye for certain giant stars.

Occultations provide an underestimate of the actual distances to stars.  The nearest star system is Alpha Centauri. It is 277000 AU (4.36 light year) away.  We have underestimated the distance to the nearest stars by only a factor of eight, which is petty good considering we needed no equipment.  Alpha Centauri is a triple star system with one member a little brighter than the Sun, one somewhat dimmer, and one too dim to worry about.  It resides in the southern sky too far away from the Zodiac to get occulted by the Moon. It would appear to blink off in two steps if it were occulted.  This is a standard way to detect double stars.

I reverse this situation to an observer of a lunar occultation on a planet in the Alpha Centauri system. Our Sun’s disk angle is 1800/277000 = 0.0065 arc second.  The occultation time would be 0.0065/0.5 = 0.013 time second is easily measurable with modern equipment but not by the eye. We could marginally revolve this angle in a grazing occultation.

In contrast to stars, planets dim slowly over tens of time seconds.  For example, the disk of Jupiter is about 40 arc seconds so it dims out in 80 time seconds.  This effect was not noted in antiquity even though planetary occultations were observed.  For example, Aristotle observed an occultation of Mars.  Alert observation would have given the disk sizes of the outer planets and the variation of the disk size of Mars as it moves closer and further from the Earth.  The crescent of Venus, however, is not obvious from occultations with the naked eye.

It turns out that naked-eye astronomers would have got an actual disk diameter estimate from observing occultations of the bright stars Aldebaran and Antares.  These stars are red giants that are more than AU wide and far more luminous than the Sun.  Their disks are 0.03 arc seconds.  This is too small to resolve in a head-on occultation, but can be resolved in a near-grazing occultation where they take about a second to dim.  If one assumed incorrectly that they are nearby sunlike stars, one would underestimate the distance of the nearest stars by a factor of 5.

Serious mostly amateur astronomers provide valuable data on occultations. The International Occultation Timing Association coordinates efforts.  Their website gives information on upcoming occultaions, nice if you do not want to wait too long outside.

http://www.lunar-occultations.com/iota

Starlight, star bright.  In his Conversation, Kepler stated that the Sun is infinitely brighter than the stars. Foscarini noted dangerously that the Sun would appear as bright as a star if viewed from far away.  Our Sun viewed from the Alpha Centauri system would be (remembering that brightness decreases with the inverse of distance squared) would be a star 1/(277000)2 or one 10 billionth as bright as the Sun viewed from the Earth.  Conversely, if we can measure the brightness of a sunlike star, we can estimate its distance to the Sun.

1690s.  The Dutch physicist Christiaan Huygens (1625-1695) makes a serious try at determining the distance to nearby stars.  He is a staunch proponent of many inhabited worlds. By now, it is a given that the Sun is a star.  He views the Sun though tiny pinholes.  He varies the disk angle of the pinhole until it is a bright as he remembers the stars are at night.  (Don’t try this unless you have a liking for guide dogs.  Huygens defocused the sunlight with a lens or dimmed it with smoked glass in addition to using the hole.) The disk angle of the pinhole gives the disk angle of the star.  He then does the same calculations that I discussed for occultations to get the distance to the star in AU.  This presumes that luminosity per surface area of the Sun is the same as that of a star.

The method has much to be desired.  We have already seen that the disk angles of stars are tiny and it is impractical to get a real pinhole far enough away.  One also needs to remember how bright the star was at night.  Huygens got about 28000 AU, which is too small by a factor of 10 for the nearest star.

The Scotsman James Gregory (1638-75) and Isaac Newton (1642-1727) used a planet for an intermediary, which can be seen at night with the stars.  Saturn is convenient as it is about as bright as the brighter stars and as it is far enough from the Sun that the distance difference between Earth to Saturn and the Sun to Saturn can be ignored in a quick calculation.  The disk angle of Saturn is 17 arc seconds (8 times 10-5 radians) and is 9.5 AU away.  It reflects about 1 billionth of the light emitted by a hemisphere of the Sun.  (This suffices to show Bruno was right that stars are much brighter than planets.)  The Sun would be this bright if it were the square root of a billion times further away than Saturn or 300,000 AU, a nice guess for Alpha Centauri.  Both Gregory and Newton used Sirius, which unlike Alpha Centauri can be seen from the British Isles.  Using more care than I did in my quick calculation, they got 83,000 and 1 million AU, respectively.  They did not know that Sirius is a lot more luminous than the Sun.  Its actual distance is 540,000 AU.

A candle makes a nice intermediary that does not require a telescope as noted by Huygens.  [see Do It Yourself Box on Distance to a Star with a Candle]

How big are the other planets and the Sun? Kepler gave the Earth a special place in his Conversation, as did Foscarini with the Sun. Their thinking must have been strongly influenced by the prevailing view of the dimensions of the solar system.  The diameter of the Earth and the distance of the Earth to the Moon were known in common units, like miles.  The relative distances of the planets from the Sun, that is the distances in AU were known.  Kepler did not know what the AU was in miles, in fact, he was the first person who tried to improve the estimates from antiquity.

It turned that Kepler’s estimate of the Sun’s distance was only the minor improvement over the Greek estimate and still low by a factor of 10.  This made the diameters of the planets too small by this factor.  Mercury, Venus, and Mars were all tiny objects smaller than the Moon.  The Earth was the same size as Jupiter and Saturn.

The Greek method is ingenious but does not work in practice (Figure 11).  When the Moon is exactly half full, the lines from the Sun to the Moon and the Moon to the observed on the Earth form a right angle.  The angle that the observer on the Earth sees is slightly less than a right angles.  As the angles of a triangle add up to 90°, the difference of that angle from a right angle is the quadrature angle Q.  One then can use similar triangles to compute the distance to the Sun in Earth-Moon distances.

Figure 11: Quadrature of the Moon gives the Sun-Earth distance H (the AU) in terms of the Earth-Moon distance D.  The angle at the Moon is a right angle.  The angle at the Earth between the Earth-Moon line D and the Earth-Sun line H is slightly less than a right angle by the quadrature angle Q.  Measurements in antiquity did show that the Sun was quite far away, but the rough surface of the Moon makes the method impractical for a precise measurement.

The actual angle is about 0.2°, too small to be resolved with the naked eye.  The Greek astronomer Aristarchus got 3° and Kepler got 2°.   They should not have incorrectly measured the Moon-Sun angle by this amount.  They missed the actual quadrature time by hours so quick reaction was not the problem.  Impatience may have been.  The waxing half-moon is in the sky with the Sun between about noon and sunset.  One sees the actual waxing quadrature only (on average) once every four months and the waning one once every four months.  The uneven surface of the Moon as seen by Galileo was a serious problem that persisted even with good telescopes.  By the late 1600s it was evident that the quadrature angle was small and could not be resolved.

A do-it-yourself project is to measure the quadrature time by naked eye or with binoculars.  Do not look up the quadrature time before you measure it so you do not bias yourself.  Check only the day when a half-moon is to occur.

It was evident that daily and geographic parallax must be used.  Mars and Venus are the nearest planets, but they move across the background of fixed stars.  The small variations from parallax were hard to separate from these motions.  The astronomers used Mars where its motion changed from retrograde to prograde (or vice versa) when these motions are as small as possible and transits of Venus across the disk of the Sun.   Martian data were easy to get but were considered unreliable.  Transits of Venus occur in pairs separated by over 100 years.

1716. Edmond Halley presents a proposal to study the next transit of Venus.  Like the return of his comet, it will not occur in his lifetime.  The proposal has a distinctly modern bent.  Here it is with my annotations in square brackets.  This translation of Halley's paper is taken from the Abridged Transactions of the Royal Society, Volume VI, pp.243-249, published in 1809.  The full text is at

http://sunearth.gsfc.nasa.gov/eclipse/transit/HalleyParallax.html

Here are NASA home pages for the 2004 and 2012 transits

http://sunearth.gsfc.nasa.gov/eclipse/OH/transit04.html

http://sunearth.gsfc.nasa.gov/eclipse/transit/venus0412.html

French and British scientists observed the transits of 1761 and 1769.  They did this when their countries were at war.  When the observers finally returned with tales of the far reaches of the Earth, the data yielded essentially the result that the AU is 150 million kilometers in modern units.

Perhaps ironically, Bruno may have guessed the right result when he visited Oxford.  The English Bishop Francis Godwin (1562-1633) wrote the adventure book “The Man in the Moone.” which was published in 1638 after his death in 1633.  Godwin was a student at Oxford during Bruno’s visit, but it unclear what he got from Bruno.  His adventurer contemplates the difficulty of the fixed stars circling the Earth each day when he is part way to the Moon:

“those same huge bodies of the fixed stars in the highest orbe, whereof divers are by themselves confessed to be more then one hundreth [hundred] times as bigge as the whole earth”  (Modern number for the Sun is 109)  (page 70)

Link to text of book:

http://e3.uci.edu/clients/bjbecker/ExploringtheCosmos/week2e.html

Update and parallax.  Actual measurements of parallax proved difficult to make.  By the late 1600s equipment was good enough to try. In 1725, James Bradley conducted a careful experiment.  He placed a fixed telescope so that a bright star would pass vertically overhead and measured its position each time it passed the north-south meridian.  (Bright stars are in fact visible with telescopes during the day.)  He found out that the star (and all others subsequently looked at) swept out an ellipse in the sky of about 40 arc seconds.  This was too large for the expected parallax and in the wrong orientation.  He had discovered stellar aberration (Figure 5).  Light travels at a finite rate, which was already known in 1725 from observing the moons of Jupiter.  The effect is much like rain hitting a car.  If it is falling straight down when the car is stopped, it will appear to fall toward the windshield when the car is moving forward and into the back windshield when it is backing.  This was conclusive, though belated, evidence that the Earth in fact circled the Sun.  Bradley obtained an accurate estimate of the speed of light from the effect.

It was thus necessary to use the apparent motion of a nearby star relative to distant background stars to measure parallax.  This was not done until 1837.  The main problems were that one needs a telescope that does not distort when it magnifies and that other stars are moving with respect to the Sun.  Halley discovered this effect called proper motion by comparing star charts from antiquity with modern star positions.  Nearby stars typically cover several AU in a year, so their proper motions against the distant background in that time are more than their parallax.  One needed to observe a star for several years to resolve parallax.

Astronomers now have a good picture of the vastness of space.  Their method uses parallax to obtain the distances of nearby stars.  They obtain objects of known brightness (standard candles) from this data and work out.  The method is basically that used by Newton and Huygens, except we now know that assuming all stars are like the Sun makes a poor standard candle.

We live in the outer part of the disk of a spiral galaxy with hundreds of billions of stars.  There are hundreds of billions of other galaxies out to distances of over 12 billion light years.  The universe is vast.  Our technology limits us to looking for life nearby, around the neighboring stars seen by Galileo and Halley.

Copyright Notice

back to top