Saturday, December 5, 2009

Where Can Venus Be?

There are several places in The Baroque Cycle where the planet Venus is mentioned. The first occurs in late winter or early spring of 1666 (page 146 of Quicksilver). At dawn, Daniel Waterhouse “found that he was walking directly toward a blazing planet, a few degrees above the western horizon, which could only be Venus.” There was “nothing before him but the Dawn Star.”

A few days later, Daniel arrived at Woolsthorpe to assist Isaac Newton in observations of Venus (pages 150-156). The house, “made of ... soft pale stone”, had a “clear sunny exposure” at its southern end, with “almost no windows there–just a couple of them, scarcely larger than gunslits, ...” As seen inside the house, “The southern half–with just a few tiny apertures to admit the plentiful sunlight–was Isaac’s: ...” “The sun was going down, and they were preparing for Venus to wheel around into the southern sky.” “Isaac had worked out during which hours of the night Venus would be shining her perfectly unidirectional light on Woolsthorpe Manor’s south wall, and he’d done it not only for tonight but for every night in the next several weeks.” “When Daniel looked, he realized that he could see not only the spectrum from Venus, but tiny, ghostly streaks of color all over the wall: the spectra cast by the stars that surrounded Venus in the southern sky.” “The earth spun and the ribbons of color migrated across the invisible wall, an inch a minute, ...”

Finally, in the early evening of 28 July 1714 (page 539 in The System of the World), Lord Ravenscar is with Lord Bolingbroke, preparing to use Bolingbroke’s telescope. Bolingbroke says, “Presently night shall fall and Venus shall shine forth.”

Let us consider whether or not Venus could have done those things at those times.

Venus, like Mercury, is an inferior planet relative to Earth, i.e., closer to the Sun. The first thing that means is that Venus, as viewed from Earth, can never get very far from the Sun. The direct result of that, is that the simplest description of its motion is 'synodic', i.e., relative to the Sun, rather than relative to the fixed stars. The plane of the orbit of Venus is inclined at an angle of about 3.4 degrees from the plane of Earth’s orbit. Therefore, Venus is never very far from the ecliptic, the path which the Sun appears to follow throughout the year. If the point of maximum excursion of Venus from the ecliptic happens to coincide with its closest approach to Earth, then it can appear to be as much as about 9 degrees north or south of the ecliptic.

To get a reasonable approximate description of Venus’s motion along the ecliptic, let us make the simplifying assumptions that Earth and Venus move in circular orbits (rather than the actual ellipses), and at constant speeds (rather than faster when closer to the Sun). We can also ignore the inclination of Venus’s orbit, so that everything of interest happens on one plane. Earth’s sidereal period of 365.25 days (i.e., relative to the stars), and Venus’s sidereal period of 224.7 days, combine to produce a synodic period of 584 days. During that period, Earth performs 1.599 revolutions, and Venus performs 2.599 revolutions, or exactly one additional revolution. Thus, the original configuration of Sun, Venus, and Earth is exactly reproduced after any integer number of synodic periods.

In evaluating the synodic description, Earth and Sun are considered fixed at a separation of one astronomical unit (AU). Venus revolves eastward (counter-clockwise as viewed from ecliptic north) in a circle of radius 0.723 AU. Its angular speed is 0.6164 degrees per day, so that it goes once around the circle in 584 days. The 'elongation' of Venus, which is its angular displacement from the Sun as viewed from Earth, can be found at any time by solving the Sun-Earth-Venus triangle.

Let us start with zero elongation, increasing eastward. Venus is then directly on the far side of the Sun from Earth ('superior conjunction'). At that time, it has its maximum velocity in elongation, at 0.259 degrees per day. The apparent angular speed decreases continuously for about 217 days, at which point the line of sight from Venus to Earth is tangential to Venus’s orbit. This is the condition of 'greatest eastward elongation', and Venus is 46.3 degrees east of the Sun. (Note that sine 46.3 degrees = 0.723, the ratio of orbit radii.) Venus’s synodic motion is very slow at this time. It spends about 24 days within 0.5 degree of greatest elongation. (That angle is the apparent width of Sun or Moon.)

After greatest eastward elongation, Venus moves westward (toward the Sun). Its speed in elongation increases continuously for about 75 days, until it is directly between Earth and Sun ('inferior conjunction'). The maximum westward speed is about 1.609 degrees per day. During this half-synodic-period of 292 days, Venus is east of the Sun, and is visible in the western sky after sunset (an 'evening star'). As viewed through a telescope, it exhibits phases, similar to those of the moon. Near superior conjunction, it appears 'full' (circular). It is 'waning gibbous' until greatest eastward elongation, when it appears 'half full'. Thereafter it is 'waning crescent', until it is 'dark' near inferior conjunction.

The second half-period of the synodic motion is essentially the mirror image of the first half-period. For about 75 days the apparent angular speed decreases continuously until Venus reaches greatest westward elongation at 46.3 degrees west of the Sun. It then starts moving eastward, until after another 217 days of continually increasing angular speed it reaches superior conjunction again. During this entire time it is visible in the eastern sky before sunrise (a 'morning star', or the “Dawn star”). Its phases are successively 'dark', 'waxing crescent', 'half full' (at greatest westward elongation), 'waxing gibbous', and 'full'.

The motion of Venus relative to the stars can be found by superimposing the Sun’s average motion (about 0.986 degrees per day eastward along the ecliptic) upon this synodic motion. The effects of the eccentricity of the orbits of Venus and Earth are to change the angles of greatest elongation by a degree or two, and to change the time intervals between unique configurations by a few days.

It is obvious that the first described appearance of Venus is quite impossible. Any planet which is about 180 degrees from the Sun must be superior to Earth, i.e., further from the Sun. That configuration is 'opposition', and it is the time at which such a planet is closest to Earth, and hence appears brightest. The two best candidate planets are Jupiter and Saturn, because they appear white, like Venus. Mars is less bright, and appears reddish rather than “blazing”. I tried online to find where the superior planets were at that time, but I gave up (perhaps too soon) on finding a free historical ephemeris. Apparently there is still so much money to be made in astrology, that the proprietors of ephemerides insist on being paid.

The description of Woolsthorpe Manor, given by Stephenson, is not a very good match to the photographs easily found in a search on that place name. (See e.g., Wikipedia.) The photos show several good-sized windows on the sunlit broad front of the house. However, there could have been extensive alterations in the intervening centuries.

The first problem with viewing the spectrum of Venus lies in getting the light through the wall of the house, as described by Stephenson. My guess is that the masonry might be eight inches thick. If the apertures were made like “gunslits”, they might be relieved at an angle of perhaps 45 degrees on each side. In such a case, light from only those directions between south-east and south-west could get in.

Normal window openings, perhaps two feet wide, would relieve the problem almost completely. Opaque panels over the sashes could darken the room. If such a panel were near the midpoint of the thickness of the wall, then an aperture in the center of the panel would admit light from any direction less than about 71 degrees from south.

It is difficult to reconcile Stephenson’s statement, about Venus wheeling around into the southern sky, with the above description of Venus’s synodic motion. If Venus is an evening star, then at sunset it is already as far south as it will ever be during that night. (The only possible exception is if greatest eastward elongation nearly coincides with winter solstice. At that high latitude (about 52.8 degrees), the sun then sets well south of west. Calculations of the azimuths of sunset and of Venus at that time are left to the interested reader.)

On the other hand, if Venus is a morning star, then sunset has nothing to do with the situation. The observers should go to bed early, so that they can get up before sunrise, when Venus is sufficiently south of east.

Newton’s calculations of Venus’s visibility on various nights were no big deal. Except close to inferior conjunction, Venus moves so sedately that times for visibility would change by only a few minutes over the course of a week.

In order to see a spectrum of Venus’s light with the unaided eye, Venus should be as bright as possible. Its brightness depends upon both the phase, which determines how much illuminated area is exposed, and the distance from Earth, which determines the solid angle subtended by that illuminated area. The maximum brightness occurs between greatest elongation and inferior conjunction, but is only slightly greater than the brightness at greatest elongation.

The really big question remains: where was Venus relative to the sun in the early spring of 1666? Fortunately, the free site presents the article “Viewing Venus in Broad Daylight”, by John Walker. It includes a calculator for dates of greatest elongations of Venus, for arbitrary years. It does not offer dates of conjunctions. The dates which span the time of interest are as follows.
1665 Sep 11```46.461 deg W
1666 Nov 29```47.473 deg E

Midway between those two dates was 1666 Apr 21. That should be within perhaps 5 days of the true date of superior conjunction. Unfortunately, that website does not identify which calendar is used for those dates. I would guess that it is Gregorian (“New Style”), which was then used in most of continental Europe. In Britain, they still used the Julian calendar (“Old Style”), and called it April 11.

I don’t know exactly when apple trees bloom in Lincolnshire, but it almost has to be rather close to April 21. Unfortunately, superior conjunction is the worst possible time to make observations of Venus. The planet is as far away from Earth as it ever gets, so that it has only about one-third its maximum brightness. It is lost in the glow of the twilight sky, until it gets far enough away from the sun. At a guess, it should be at least 10 degrees away, in order to give any useful viewing time. That would take about six weeks after conjunction. As the final blow, Venus would be very close to directly west at sunset. Its light would not shine on the south wall of Woolsthorpe Manor at all, during hours of darkness.

Finally, the dates of greatest elongation which span midsummer of 1714 are as follows.
1713 Aug 29```45.459 deg W
1714 Nov 16```47.473 deg E
The rather close match of these dates to the previous case arises because a difference of 30 synodic periods amounts to just less than 48 Earth years. The superior conjunction for this case happened close to 1714 April 8 (Gregorian).

The evening in question [28 July (Julian) or 8 August (Gregorian)] was 122 days later, and Venus’s elongation would have been about 30.5 degrees Eastward. It was indeed an evening star on that date, and would have been clearly visible in the western sky after sunset. (Unfortunately, the seeing was probably very bad after all the bonfires were started.)

Thus Stephenson put Venus in the right place one time out of three.

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