Signs in the Heavens

I'm looking for [conjunctions, sextiles, trines, squares, quintiles] of [Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto, North Node, South Node, Trans-plutonian, ...]

• Before 1800, Astronomy and Astrology are one discipline.

What are you surveying here?

• The First Heaven conventionally covers the Earth's atmosphere. I will also include the Earth's magnetosphere in this category, and related phenomena.
• The Second Heaven conventionally includes everything beyond the Earth's atmosphere. I will define this to be the vacuum of space, and anything normally separated from Earth by the vacuum of space.
• The Third Heaven is not accessible to even Science At The Limit. I quote the Apostle Paul:

It is doubtless not profitable for me to boast. I will come to visions and revelations of the Lord: I know a man in Christ who fourteen years ago -- whether in the body I do not know, or whether out of the body I do not know, God knows -- such a one was caught up to the third heaven. And I know such a man -- whether in the body or out of the body, I do not know, God knows -- how he was caught up into Paradise and heard inexpressible words, which it is not lawful for a man to utter. Of such a one I will boast; yet of myself will I not boast, except in my infirmities. For though I might desire to boast, I will not be a fool; for I will speak the truth. But I refrain; lest anyone should think of me above what he sees me to be or hears from me.

[II Cor. 12:1..6, NKJV]

General resources

EXERCISE: try to figure out when I'm paraphrasing the resources I mention above -- and which one.

Most other exercises, below, are explicitly worked in one of the above general resources.

The First Heaven

Heat index charts

• UNISYS Temperature is in F°. Either dewpoint or relative humidity can be used as the second parameter.

Wind chill charts

• UNISYS Temperature is in F°. Wind speed is in knots (nautical miles per hour).

The height of the tropopause varies from about 10 miles at the equator to about 5 miles at the poles, and is about 7 miles high in the mid-latitudes (circa 45° north or south). These heights are measured from sea level.

The stratosphere (the layer Léon-Philippe Teisserenc de Bort's balloons reached) extends to about 30 miles above sea level. Temperatures generally range from -40°F to -100°F. Except for the lower reaches of the jet stream, the stratosphere is virtually windless.

It should be noted that ozone (O₃) has measurable concentrations from about 6 miles above sea level to the top of the stratosphere. Most ozone formation happens 20-30 km above sea level, as a side-effect of absorption of UV radiation (particularly UV-C) by oxygen (O₂); the temperature in this layer climbs from -60°F to 130°F in less than 200 feet.

The mesosphere extends from the end of the stratosphere to about 50 miles above sea level. Temperatures generally decrease with height, attaining a low of -225°F in the summer. The mesosphere is warmer in the winter than in the summer. This is conjectured to be due to a heat-exchange effect. Most meteors burn up in the mesosphere. The upper range of the jet stream is in the lower mesosphere.

I would naïvely conjecture that increasing the "thermal inertia" of the atmosphere by augmenting carbon dioxide (CO₂) should reduce the temperature in the mesosphere, since summer already reduces the temperature of the mesophere compared to winter. At least one U.S. NASA simulation (I'd have to search archives, but it was clipped to MyCNN during the year 1999 or 2000) has confirmed this at an altitude of 40 miles above sea level.

Note: the popular literature calls 'increasing the "thermal inertia" of the atmosphere by augmenting carbon dioxide' the "greenhouse effect". This is not a simple heating up; rather, all weather systems have their thermal energy increased, permitting increased severity.

The jet stream(s) are a thin (typically 1-2 miles high), wide (typically 180-300 miles wide) band of wind (ranging from 60 miles per hour to an extreme of about 290 miles per hour, but usually not faster than 150 miles per hour).

The thermosphere extends from the top of the mesosphere, to about 180 miles above sea level. Temperatures in this region can fluctuate from highs of about 3,600°F to lows of about 2,600°F.

The exosphere extends from the top of the mesosphere into the vacuum of space. The latter boundary is very dubious, but may be safely said to be somewhere between 300 and 900 miles above sea level. The main components here are hydrogen, helium, and oxygen (often monoatomic). Some of these atoms attain escape velocity and leave the Earth indefinitely.

The above was abstracted from, among other sources, "It's Raining Frogs and Fishes: Four Seasons of Natural Phenomena and Oddities of the Sky" by Jerry Dennis, published by Harper-Collins.

This section is not implemented yet.

The Second Heaven

• Human vision cannot effectively use parallax information to judge distance beyond 'optical infinity' (about 500 feet). [That is, for those 97-98%(?) of the people who have true stereoscopic vision.]
• At the distance scales normally used in astronomy (certainly the moon and beyond), precision measurements are required to detect parallax at all. For instance, using a specific diameter of the Earth's orbit as a baseline, Proxima Centauri's apparent position is shifted by about 1.54 arc-seconds when viewed from opposite endpoints of this diameter. This yields a reported parallax of 0.77 arc-seconds. (Proxima Centauri is 11th magnitude. The nearest visible star is Alpha Centauri -- 4.4 light-years away, rather than Proxima Centauri's distance of 4.2 light-years).

This means that Earth-based observations need only specify two out of the three 3-d spherical coordinates: the angle theta (angular offset from the celestial zero north-south meridian), and the angle phi (angular offset from the celestial equator). The radius r cannot be measured directly. Alternatively, we could use offsets from the observer's meridian and horizon (defined before).

EXERCISE: Since the sun and stars rise in the east and set in the west, which way does the Earth rotate?

There is at least one low-tech way to know that the Earth is spherical without extensive travel: lunar eclipses. Pythagoras (ancient Greece) noted that in a lunar eclipse, the shadow of the Earth on the Moon is always curved, and always has the same degree of curvature. The only type of object whose shadow has this property, regardless of orientation, is a sphere. [In practice, this means that within the observational error above, the Earth is a sphere. This means that the difference in the polar diameter, versus equatorial diameter(s), of the Earth does not affect the visible curvature of the shadow of the Earth. There should be a slight difference because a certain amount of gravitational acceleration+tension forces at the equator is required to anchor the Earth's crust to the Earth. This difference is measurable.]

Eratosthenes (approximately 276BC-195BC) took advantage of his position as the Librarian at the great Museum in Alexandria to estimate the circumference of the Earth from this. He noted that at summer solstice, there was no visible shadow at noon in Alexandria (use a deep well to check this), but there was a shadow at Syene at summer solstice. The angle of this shadow was just over 7°. From this, he estimated that the distance between Alexandria and Syene was 1/50th of a great circle. Since the distance between the two cities was "5,000 stadia", the circumference of the Earth was "250,000 stadia". Unfortunately, we don't have a clear idea what his length unit was in our units. Conventional approximations suggest his figure was about 47,000 kilometers/29,000 miles (two significant digits), versus a modern five-significant-digit approximation of 40,074 kilometers/24,902 miles. [Abbreviated forms of these: "47,000 km/29,000 mi (two significant digits), versus a modern five-significant-digit approximation of 40,074 km/24,902 mi". Yes, the period is deliberately not used in the abbreviation for miles, contravening the usual grammatical rule for abbreviations that use direct truncations.]

You might try clicking on the left or right third of the celestial sphere illustration, below--to rotate it.

The north and south celestial poles [following Strobel, I will use the acronyms NCP and SCP respectively when I don't need literary parallelism] are defined by the Earth's rotation axis. That is:

• The Earth's rotation axis intersects the celestial sphere, directly over the north pole, at the north celestial pole. It is visible in the northern hemisphere, and on the horizon on the equator.
• The Earth's rotation axis also intersects the celestial sphere, directly over the south pole, at the south celestial pole. It is visible in the southern hemisphere, and on the horizon at the equator.

• At the North and South Poles, due east and due west are meaningless, so we know the above description breaks down. The celestial equator is the horizon at the North and South Poles.

EXERCISE: Explain why the above is true, in terms of the Earth's rotation and the way we are using spherical coordinates to describe the celestial sphere. [This may be interpreted as either a math or an essay question, depending on one's talents. Do it the easy way for you.]

EXERCISE: What is the angle of the stars' path in the sky relative to the horizon? [Yes, this depends on your latitude. HINTS: Interpret 'parallel to the horizon' as an angle of 0°; this will happen at the North and South Poles. Also, the celestial north and south poles are on the horizon on the Earth's equator.]

We are now ready to define celestial coordinate systems. There are two commonly used: altitude-azimuth, and equatorial. The first is defined in terms of one's observational point, while the second is (almost) fixed relative to the stars.

Altitude-azimuth system:

• Altitude: angle above the horizon. [A "negative" angle is below the horizon, thus not visible.]
• Azimuth: clockwise angle from geographic north.
  • Geographic north is not magnetic north, just very close (most places on Earth, anyway). In the northern hemisphere, Polaris is the (current) North Star, and will be at practically exactly the celestial north pole in the year 2007.

Equatorial:

• Right ascension: the time it takes to rotate the object's position to the local meridian, measured from when the vernal equinox is at the local meridian. This is measured in hours, minutes, and seconds; 24 hours = 360°, and the other two definitions are usual [60 minutes=1 hour, 60 seconds=1 minute]. EXERCISE: work the coordinate conversion table.
• Declination: the angle the object's position makes with the celestial equator. The North celestial hemisphere has positive angles, the south celestial hemisphere has negative angles.

I am defining spherical coordinates as radius r, theta 0°-360° counterclockwise, and phi 0° to 180° from 'vertical up' [analog north pole, zenith] to 'vertical down' [analog south pole, nadir].

• I know that another common way of defining phi is 90° to -90° from 'vertical up' to 'vertical down'; however, this can do strange things to volume and area calculations. [A naive setup can give negative volumes and/or areas that must be corrected with absolute values.]
• [Skip the math!] THIS COMMENT ASSUMES FAMILIARITY WITH ITERATED INTEGRALS. Things work out correctly for the volume integral of a sphere, however. We work all the integrals in radians. For instance, under my preferred definition:
  • volume of sphere of radius r = ± r ²sin(f) df dq dr
  • = ±2p r ² sin(f) df dr
  • = ±2pr ² [-cos(f) ] dr
  • = ±2pr ² [-cos(p)+cos(0)] dr
  • = ±2pr ² 2 dr
  • = 4p r ² dr
• With the other common definition, we get:
  • volume of sphere of radius r = ± r ²cos(f) df dq dr
  • = ±2p r ² cos(f) df dr
  • = ±2pr ² [sin(f) ] dr
  • = ±2pr ² 2sin(p/2) dr
  • = ±2pr ² 2 dr
  • = 4p r ² dr
• At this point:
  • volume of sphere of radius r = 4p[r ³/3 ]
  • = (4p/3) r ³

• NOTE: The First Coming of Christ was with a well-defined Pole Star [Kochab].

Mr. Strobel has a very elegant explanation of which stars are above the horizon longer, or shorter, than 12 hours per day.

Circumpolar stars are those stars that either never rise above the horizon, or never set below the horizon.

• On the equator, the only possible places for circumpolar stars are the north and south celestial poles.
• In the northern hemisphere, stars with a celestial latitude at least as far north as 90°-observers latitude are circumpolar, and always visible. Stars with a celestial latitude as least as far south as 90°-observer's latitude are circumpolar, and never visible. [In the southern hemisphere, reverse the visibility classifications: "north of..." is never visible, and "south of..." is always visible.].

Observational terms that have nothing to do with the 'fixed stars' (and thus will move relative to the fixed stars as the Earth rotates):

• Zenith: the point directly overhead. It is 90° above the horizon.
• Nadir: the point directly underneath. It is 90° below the horizon.
• Meridian: the arc through the north point on the horizon, and the south point on the horizon, through the zenith.

EXERCISE: What altitude does the north celestial pole have in the northern hemisphere? [Alternatively, south celestial pole in the southern hemisphere; however, the answer will be the same. Again, this depends on your latitude.]

The solar day, at a given location, may be defined by the time between consecutive instances of the Sun being on the local meridian.

NOTE: the Earth's axis is tilted, relative to the Earth's orbital plane, by about 23.5°. I will omit the word 'about' in further usage, but it should be kept in mind.

And now, some terms that have to do with the Earth's geography:

• The Tropic of Cancer: the latitude circle of 23.5° N [the Earth's axis tilt N]. This is defined by the maximum northward celestial latitude the Sun attains during the year (on the summer solstice for the Northern hemisphere, winter solstice for the Southern hemisphere).
• The Tropic of Capricorn: the latitude circle of 23.5° S [the Earth's axis tilt S] This is defined by the maximum northward celestial latitude the Sun attains during the year (on the winter solstice for the Northern hemisphere, summer solstice for the Southern hemisphere).
• The Arctic Circle: the latitude circle of 66.5° N [90° - the Earth's axis tilt N]
• The Antarctic Circle: the latitude circle of 66.5° S [90° - the Earth's axis tilt S]

Areas north of the Arctic circle, or south of the Antarctic circle, have some calendar days where the sun is above the horizon 24 hours per day. This has prompted the term "land of the midnight sun".

Now, the Sun appears to drift eastwards relative, relative to the stars, over a year's time. The path of its drift is the projection of the Earth's orbit onto the sky; we call it the ecliptic. Since the Earth's axis is tilted by 23.5° relative to the Earth's orbital plane, the ecliptic is tilted by 23.5° relative to the celestial equator.

The ecliptic intersects the celestial equator at two points: the vernal (spring) and autumnal (fall) equinoxes. [Yes, this is named for northern hemisphere seasons, for historical reasons.]

• The sun crosses from the south celestial hemisphere to the north celestial hemisphere at the vernal equinox. This is about March 21 in the Gregorian calendar.
• The sun crosses from the north celestial hemisphere to the south celestial hemisphere at the autumnal equinox. This is about September 22 in the Gregorian calendar.
• For most of the Earth, day and night are both 12 hours long when the sun is at the equinoxes. Areas with a latitude more extreme than 89.75° are the exception:
  • On the vernal equinox, the sun is rising (very slowly) near the north pole, and setting (very slowly) near the south pole.
  • On the autumnal equinox, the sun is setting (very slowly) near the north pole, and rising (very slowly) near the south pole.

North of the Arctic Circle, or south of the Antarctic Circle, if the Sun's celestial latitude is within 0.25° of 90°-the observer's latitude, then the Sun is either rising (or setting) the entire 24 hours of the calendar day.

Most of the above is covered by Mr. Stroebel's notes.

Tycho Brahe (1546-1601 AD/CE) started a multi-year program to acquire accurate observational data on planetary positions to test the Ptolemaic description of apparent planetary motions in the sky. This started with the construction of an observatory capable of measurements accurate to one arc-minute (as opposed to ten arc-minutes anywhere else in Europe) at the Danish island of Hven in 1576.

Note: Using the Earth's rotation as a baseline, Tycho Brahe could triangulate out to at least cot(1.5')·cos(latitude of Hven)·(Earth's radius). Other European observatories were limited to cot(15')·cos(latitude of observatory)·(Earth's radius). Numerically, (bounding the cosine below by 0.6), this works out to a Tychonic triangulation range of at least 2750 Earth diameters, as opposed to a general European triangulation range of at least 275 Earth diameters. Similar scaling factors are applicable to using the Earth's orbit as a baseline. (This is inconceivable at the time: Copernicus will use Kepler's work to reconsider a heliocentric solar system, seriously proposed by the Greek Aristarchus by 280 BC.) The European range is quite enough to triangulate the distance of the moon. [Hipparchus, a Greek astronomer based at the island of Rhodes, estimated the distance of the Moon, from the Earth, as 67.5 Earth radii simply by combining the apparent speed of the Moon across the sky with the fact that the Moon exactly covers the Sun during a total solar eclipse. It is possible to do much better than this.]

Tycho Brahe attempted to measure the change in angular position of a supernova (as we now call it) that appeared in the constellation Cassiopeia on November 11, 1572. No change was measurable, showing that the supernova was almost certainly both not moving signficantly, and beyond Tycho Brahe's triangulation range. [Since a common belief (due to Aristotle, a very unempirical Greek popular with the Roman Catholic Church then) at the time was that supernovae ["new stars"] were atmospheric phenomena because they changed, this failure to measure was academically significant.] Two bright comets in 1577 and 1588 both were measured as beyond the Moon (causing similar academic difficulties, for the same reason).

Johannes Kepler (1571-1630 CE/AD) was hired onto Tycho Brahe's staff in 1600 with the explicit job of retrofitting the Ptolemaic model to 20 years of observatory data on the apparent position of the planet Mars. However, Kepler was unable to do so -- the best fit he found had errors of eight arc-seconds, thus invalidated by the one arc-second accuracy of Tycho's observatory at Hven. He empirically constructed the three laws named after him:

• [1609: Astronomia Nova i.e. Commentaries on the Motions of Mars]
  Kepler's First Law: Planetary orbits are ellipses, with the Sun at one focus.
• [1609: Astronomia Nova i.e. Commentaries on the Motions of Mars]
  Kepler's Second Law: A line drawn between a planet, and the Sun, sweeps out equal areas in equal times.
• [1619: The Harmony of the World]
  Kepler's Third Law:
  • (Planet #1's sidereal period/Planet #2's sidereal period)²
  • =(Planet #1's average distance from sun/Planet #2's average distance from Sun)³
  • [NOTE: above is as published. I need to check whether the correct version depends on semimajor axis (average of perihelion [minimum] and aphelion [maximum] distances from the Sun).]

Much of the above is from "The Illustrated Encyclopedia of the Universe", ©2001 The Foundry Media Company Ltd. The exact publication dates and methods for Kepler's laws is from "Mathematical Thought from Ancient to Modern Times", ©1972 by Morris Kline.

• Apparent angular diameter: about 30'

A (relatively) simple way for us to replicate this measurement uses a tool Aristarchus did not have: an accurate clock, allowing one to measure time down to one second. [At least, that's what occurred to us when I talked with George Fisher.] By exactly measuring when the Moon attains first/third quarter, and also exactly measuring when solar noon is, we can obtain an angular displacement between the Moon at first/third quarter and the Sun at solar noon. [4 minutes of time equals 1° of rotation. An accuracy of 2 seconds suffices to measure down to 1 minute of accuracy.] We then apply a correction for the Sun's movement on the ecliptic to obtain where the Sun was when the Moon was measured at first/third quarter. Proceed from there, being certain to use intervals for calculation.

The above is from "Mathematical Thought From Ancient to Modern Times" by Morris Kline, Vol. 1, [section 4 "Greek Mathematical Astronomy" of chapter 7 "The Greek Rationalization of Nature".]

The Moon
Clementine probe summary from the U.S. Navy

• Apparent angular diameter: about 30'

Mars
Weather from Malin Space Science Systems
Atlas from Malin Space Science Systems
U.S. NASA's exploration project and landing sites

This section is not completely implemented yet.

Meteors and Asteroids
Meteor shower calendar
Trojan asteroid enumeration
AA29: Almost in Earth's orbit.

This section is not completely implemented yet.

Let's start with the Hertzsprung-Russell diagram, a method of graphing stars by effective temperature (the temperature which the star is actually radiating at, when thought of as a black-body) against any of luminosity, apparent magnitude, or absolute magnitude. Historically, temperature is plotted as decreasing from left to right. The full temperature range that can be graphed includes 2,000K° to 50,000K°. [An example diagram will have to wait until I obtain/construct one of sufficient quality.]

It is possible to estimate the ratio of the distances of two given star clusters from a vantage point (the only useful example being the solar system) using the main sequence. Since stars within a cluster are roughly at the same distance from Earth, it is possible to chart the apparent magnitude (possibly after correcting for absorption by interstellar gas) of the stars on the same graph. The more distant cluster is fainter, so its main sequence will be vertically lower on the diagram. Since the intensity of light falls off as the square of distance, the downward shift in apparent magnitude directly measures the factor by which the more distant cluster is further away from Earth than the nearer cluster.

Apparent magnitude measures how intense the (visible) light from a star is. It is calibrated on a logarithmic scale so that a magnitude 1 star is exactly 100 times as bright in intensity as a magnitude 6 star.

Absolute magnitude is a statement of how intense the (visible) light from a star would be if the star was 10 parsecs distant from Earth (or other vantage point).

A parsec is defined as the distance that a star would have to be at to exhibit one arc-second of parallax from opposite points on the Earth's orbit. That is: think of the star as being at the vertex of an isosceles triangle opposite the side which is a diameter of the Earth's orbit. When that vertex is two seconds, the distance from the Sun to the star is one parsec. In practice, a decent approximation for a parsec is 3.26 light-years, where a light-year is the distance that light travels in a year.

This suggests that there should be a formula, using a base 10 logarithm, converting luminosity to absolute magnitude. This formula should be of the form

where b is to be determined empirically for a given system of measuring units.

If we define the Sun's current luminosity [as of 1900-2000 CE/AD] to be 1, then b is simply the estimated absolute magnitude of the Sun : 26.74 . In contrast, the apparent magnitude of the Sun is -4.83. This suggests that we are receiving approximately 4.24·10¹² times as much energy from the Sun as if it were at ten parsecs. Since radiation intensity falls off as the square of the distance, this suggests that 10 parsecs is approximately 2.06·10⁶ times as long as 1 AU (the average orbital distance of the Earth from the Sun).

Let's see if this checks. A parsec (by definition) is cot(1") AU. Numerically, this is 2.062·10⁵ AU (to four significant digits), so 10 parsecs is 2.062·10⁶ AU. Apparently, the limiting factor in this exercise is how precisely we (or the astronomers) can measure the apparent magnitude of the Sun.

NOTE: I am using the historical sign convention for magnitudes: increasing the magnitude number increases the faintness of the star. Some modern references will do it the other way around. To convert, negate the magnitude number.

We need to get this straight now: otherwise, any discussion of the life cycle of stars (from initial formation to old age and death) will have to be classed with speculative thinking, comparable with Aristotle's statements about supernovae and comets.

Now, let's review how stars could form from what is actually observed (telescopically) around us. Between stars, there are clouds of dark, cold gas and dust that are normally visible only when illuminated by stars. The illuminated clouds (and, by generalization, those we don't see) are predominantly hydrogen gas [H₂]. Other molecules, of varying complexity, are also found in these clouds. We call these molecular clouds. The largest examples, giant molecular clouds, may have a total mass of at least a million solar masses.

Now, "dark, cold gas and dust" suggests that the Boltzmann speed distribution for particles (or its quantum-corrected variants) applies. Being a large cloud, gravity is significant. If a sufficient fraction of the cloud is caught in a vaguely central gravity field whose escape velocity exceeds that of most of the gas particles in the cloud...it collapses inwards. [This was first noted, in 1902, by Sir James Jeans (1877-1946). He published analytic criteria for the initiation of this process.]

Now, let's consider the initial stages of the collapse of a molecular gas cloud. It is thinner than many terrestrial vacuums, so the ideal gas laws apply (even though it is very cold). There will be some molecular kinetic energy acquired from the cloud falling into itself, beyond the usual ideal gas law tradeoffs between volume (decreasing), pressure, and temperature. Also, total angular momentum is conserved...so even if the total angular momentum is immeasurable as a molecular gas cloud, it will eventually cause the cloud to visibly spin.

Eventually (simulation: 100,000 years, give or take a few tens of thousands of years?), the collapsed molecular gas cloud will change appearance. We then call it a protostar. Protostars are graphable on a Hertzsprung-Russell diagram -- they are large, luminous, and cool (compared to stars).

At this point, it is critical how much mass ends up in the would-be star (rather than its associated nebula). Less than 8% or so of a solar mass (80 times Jupiter's mass) is fatal: hydrogen fusion does not initiate, and the result is called a brown dwarf. [Astronomical pluralization differs from mythological pluralization: Seven brown dwarfs in the sky, vs. the fairy tale "Snow White and the Seven Dwarves".]

About half of all known stars are thought to be in binary systems. [At least, this is true within the range in which we can resolve binary stars.] It is possible for a binary system to be one entry in a larger "binary" system.

A stable triple star system is theoretically possible, with very restrictive conditions...starting with all three members being within about 1% of each other in mass, pairwise. The corresponding orbit is a "figure eight": 8 . See

for more details. It is speculated that the incidence of these triple star systems ranges from 1 per galaxy to 1 in the entire universe.

Much of the above is from "The Illustrated Encyclopedia of the Universe", ©2001 The Foundry Media Company Ltd.

This section is not implemented yet.

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