User:Evensteven/drafts

Some first thoughts
Brother of (Simon) Peter, Andrew is the "first-called" because Jesus called the brothers to discipleship first among the twelve. Just as the church at Rome is traditionally the "see of Peter", the episcopal office of the Pope, the church at Constantinople is traditionally the "see of Andrew", the episcopal office of the Ecumenical Patriarch. While Andrew himself could not have served as bishop of Constantinople, whose construction only began in the early fourth century, the city was built as the new seat of government for the eastern Roman Empire, which had grown impossibly large for effective governance from Rome alone. It was this immediate prominence of Constantinople within the empire that lent much weight to the new bishopric seated there, and for that reason it quickly became second in honor to Rome, displacing Alexandria which then became third. Thus, at that time the church also saw it as fitting that the sees of the two primary seats of government be honored by the two first-called brothers among the apostles, and Constantinople also came to be the traditional see of Andrew.

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Researched materials

 * Principle source: "Lives", . Chapter on St. Andrew, pp52-71.
 * Commemoration: Nov 30.
 * Son of a Jew named Jonah, brother of the Apostle Peter; native of Bethsaida in Galilee.
 * Preferring virginity, he declined to marry. Left home to become a disciple of Forerunner John, who was preaching repentance by the Jordan.
 * When John identified Jesus as the Lamb of God, Andrew and another disciple of John (thought to be the Apostle John the Evangelist) left John and followed Jesus. Andrew sought out his brother Simon Peter to tell him he had found the Christ , and brought him to Jesus.
 * Afterwards, while Andrew and Peter were fishing in the Sea of Galilee, Jesus called them both to be his disciples.
 * Andrew is thus called the "first-called" because he became a follower and disciple of Jesus before all others.

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History
Parallax is one of the oldest techniques for determining astronomical distances, having even been employed by the ancients in attempts to find the distances to the moon and sun. Stars though, so very much more distant, appear as mere points of light. Only in recent decades have the largest and nearest of them been observable as discs. Only the nearer stars exhibit a large enough parallax shift over time to be perceptible, and then the angles involved are miniscule. They were not noticeable at all until the late 18th century, but attempts to measure them then began in hopes of proving stellar distances.

The first successful result was produced in 1838 by German astronomer Friedrich Wilhelm Bessel, who calculated from his direct parallax measurements the distance to 61 Cygni, equivalent to three and a half parsecs (though the unit had not yet been defined). This was astronomers' first such step outside the Solar System, establishing the lowest rung of what has since become known as the cosmic distance ladder, the progressive series of techniques that now identify ever larger scales of distance in the universe. Development of parallax techniques occupied astronomers for the next 75 years and more as they developed catalogues of the nearer stellar distances.

The parsec unit must have been defined and in use before 1913, for it was then that the term parsec itself was first mentioned in an astronomical publication, when Astronomer Royal Frank Watson Dyson expressed his concern for the need of a name for that unit of distance. He proposed the name astron, but mentioned that Carl Charlier had suggested siriometer and Herbert Hall Turner had proposed parsec. It was Turner's name that stuck.

Measurements of parallax are still highly useful today for this range of distances, becoming ever more precise, so the parsec also remains firmly established within the star catalogues and current astronomical processes.

Definition


The apparent shift of position in a nearby star against the (apparently) immobile background is largest when the vantage points from which the observations are made are farthest apart, especially important since stellar parallaxes are so tiny. Until the advent of spacecraft, the best humans could do was observe a star's apparent position at intervals of about six months' time, when the Earth is on opposite sides of the Sun in its orbit. Even now it would be prohibitively expensive to send telescopes or other such equipment beyond Earth orbit, and rocket payload sizes are limiting. So the maximum baseline for parallax measurement is still, practically speaking, limited to about 2 astronomical units, the distance across Earth's orbit.

For purposes of distance calculation, however, we depend on trigonometry, where it is convenient to use the properties of right triangles. Halving the maximum possible angle observations (once made), and calling that "the parallax" of the star, allows us to consider its distance in terms of a right triangle whose right-angle vertex is at the center of the Sun. Another vertex is placed at the center of the Earth, one au distant, connected along the short leg of the triangle, half the baseline for the angle measurement. The tiny parallax angle is formed at the third vertex, at the center of the star. The star's distance to Earth is along the hypotenuse of the triangle; its distance to the Sun is the triangle's long leg. Those distances are nearly identical because the short leg is so relatively small.

We do the same thing to define the parsec, though we now define its "parallax" angle to be one arcsecond. No observation is necessary for the definition. The baseline leg, one au long, the right angle, and the parallax angle, are sufficient (as proven in geometry) to completely define the shape and size of the triangle. We call the length of the long leg one parsec, but the relationship of the parsec to other measurement units must be calculated. Trigonometry tells us how to calculate the parsec in terms of the au: take the cotangent of the parallax angle, one arcsecond, which gives approximately 206 264.81 au.

While the cotangent of a star's parallax angle would still give its distance in au, even without parsecs, calculating cotangents of such small angles can be challenging, even for computers, and the results would generally yield millions of au. The convenience of parsecs to astronomers lies in the smaller numbers of units needed to express distances, and in much-simplified calculation. The latter results from a mathematical approximation that can be substituted for cotangents, because of a trigonometric property of skinny triangles caused by the small size of the parallax angle. Even the mathematical approximation is far more accurate than we can obtain for a parallax measurement. A stellar distance in parsecs turns out to be simply the reciprocal of its parallax angle in arcseconds. Thus, the parallax for Proxima Centauri, 0.7687 arcsec, yields its distance, 1 / 0.7687 = 1.3009 parsecs, or about 268 330 au, the approximate cotangent of 0.7687 arcsec.



In the diagram above (not to scale), S represents the Sun, and E the Earth at one point in its orbit. Thus the distance ES is one astronomical unit (AU). The angle SDE is one arcsecond ($206,264.81$ of a degree) so by definition D is a point in space at a distance of one parsec from the Sun. By trigonometry, the distance SD is


 * $$SD = \frac{\mathrm{ES}}{\tan 1^{\prime\prime}}$$

Using the small-angle approximation, by which the sine (and, hence, the tangent) of an extremely small angle is essentially equal to the angle itself (in radians),


 * $$SD \approx \frac{\mathrm{ES}}{1^{\prime\prime}} = \frac{1 \, \mbox{AU}}{(\tfrac{1}{60 \times 60} \times \tfrac{\pi}{180})} = \frac{648\,000}{\pi} \, \mbox{AU} \approx 206\,264.81 \mbox{ AU} .$$

Since the astronomical unit is defined to be 149 597  870  700 metres, the following can be calculated.




 * rowspan=5 valign=top|1 parsec
 * ≈ $1/undefined$ astronomical units
 * ≈ $149,597,870,700$ metres
 * ≈ $206,264.81$ trillion miles
 * ≈ $3.086$ light years
 * }
 * ≈ $30.857$ trillion miles
 * ≈ $19.174$ light years
 * }
 * }

A corollary is that 1 parsec is also the distance from which a disc with a diameter of 1 AU must be viewed for it to have an angular diameter of 1 arcsecond (by placing the observer at D and a diameter of the disc on ES).