Astronomy

Meta: Writing this note during week one: My plan is to complete week one (i.e. first) and then try to use a holistic perspective to see if I can get some sort of MM-type [Mind Map] notes going. The final goal by the end of this course will be to have an intellectual framework for astronomy.

2: The Celestial Sphere

• Modern astronomy recognizes 88 constellations as areas (with defined boundaries) that cover the sky, and can be used as points of reference (e.g. alpha Aries is the primary star within that constellation).
• Stars located in terms of a celestial sphere, being an imaginary enormous sphere with the Earth at its centre.
• Declination = Measure of celestial latitude (marked as lines of declination), measured as degrees from celestial equator (corresponding to the Earth’s equator).
• Right ascension = Measure of celestial longitude (marked as hour circles), measured in hours to east from the prime celestial meridian (being at right angle to the vernal equinox; the place where the sun passes celestial equator moving south-north, in the constellation Pisces*. Vernal equinox is the northern hemisphere’s spring equinox). It corresponds with the rotation of the earth, such that an object at 30-degrees longitude is measured as 2 hours RA, that being how far the sky rotates in that time.
  • *This had historically been called the first point of Aries, but due to the procession of equinoxes, the position has moved to within Pisces.
  • Cf. A terrestrial meridian is a line of longitude.

[Via The Celestial Sphere @ Astronomy WI]

• The path traced by the sun over a 1-year cycle is called the ecliptic – the height of the sun (maximum at summer) moves W-E by 1-degree/day. (For heaps more see Wikipedia article).
• The sidereal month for the moon is the time taken to rotate the Earth relative to fixed stars (27.3 days), which is shorter than the synodic month, which is the time between identical phases of the moon (29.5 days), due to the displacement of the Earth over the period requiring further movement to achieve the same relative geometries. (More at Wikipedia article).
• [Might read more from this source, but am realizing that at this stage of the course, these facts are not yet directly relevant]

3: The Local View

• A star’s position above the horizon is given as altitude in degrees above the horizon, which can also be measured in relation to my zenith (directly overhead) altitude (equals 90 degrees minus altitude).
• The celestial north/south poles correspond with the zenith above the respective poles of the Earth’s axis of rotation.
• Position around the horizon is given relative to the local meridian* (in terms of the north/south line it offers), given in degrees. Often given as azimuth, which is the angle from north.
  • *The local meridian is a circle passing through the celestial poles and the zenith (i.e. of my location).
  • *Cf. For terrestrial geography, a meridian is a line of latitude (i.e. connecting the poles along the planet’s surface).
• It can be shown (by drawing triangles on an appropriately labelled celestial sphere) that one’s latitude will correspond with the altitude of the visible pole in the sky (e.g. at the north pole I am at 90 degrees latitude, and that is the position of the celestial north pole in my sky; similarly if I am at latitude 35 degrees, that is the hight of my CNP). Thus can measure position.

[Via Astronomy without a telescope @ Astronomy Note]

• The stars rotate in the sky in a path that is parallel to the celestial equator.
• The passing of a star through the local meridian marks the highest altitude it reaches during that day-night cycle.
• The angle of stars’ rotation in the sky can be measured as an angle relative to the horizon, which is obviously the same for the entire sky at any location. It can be given as equals 90-degrees minus the observer’s latitude. Thus (as said) at the poles the stars never set, and the angle becomes steeper the closer to the equator.
• During daylight, the meridian separates the morning (AM, Latin. antemeridiem) and afternoon (PM, post-meridiem) positions of the sun. At noon, in the southern hemisphere, the sun is due north (and vice versa).
• As I move north from the south pole, with every degree: (1) the position of the SCP moves 1-degree away from (i.e. my) zenith southwards (nb. the SCP’s position on the celestial sphere, which is theoretically infinite in diameter, is easily visualized by giving it a proxy, e.g. using a star); (2) the highest point of the celestial equator moves 1-degree higher from the northern horizon.
• Nb. It follows that at the Earth’s equator, the SNP is on the southern horizon, the celestial equator is at 90-degrees, and thus directly overhead E-W, and the stars rise/set perpendicular to the horizon (i.e. straight up/down). This last point means that we see half of the stars’ full rotation around the Earth, similar to the rotation of the sun we see during the day.

4: Sidereal Time

• Stars around the celestial pole that are circumpolar, those in the midst of summer never set and rotate around the pole, and in the midst of winter at the opposite pole, they never rise. Stars in between the two circumpolar regions rise, move west, and set.
• Note that our zenith‘s declination is equal to our latitude. And vice versa.
  • In other words, the position of the zenith’s degrees perpendicular, i.e. declination, away from the celestial equator, is the same as our position in degrees perpendicular, i.e. latitude, away from the terrestrial equator. This is easy to see in the case of the north pole, when I’m at 90-degrees latitude north, and my zenith is at 90-degrees declination, i.e. at the celestial pole).
• Note that our zenith‘s right ascension (in other words the celestial, corresponding to the terrestrial, meridian above our location) is thus also our sidereal time.
• Sidereal time is the name of the prime meridian (cf. terrestrial) which corresponds to the local celestial meridian.
  • Nb. Proceeds with time. Similarly moves forward by moving east.
  • Thus 24-sidereal hours correspond to a 360-degree rotation of the Earth around its axis.
  • Thus can measure time, as in 1-(sidereal) hour, the celestial sphere shifts by 1-hour of right ascension.
  • Thus the position of stars in my sky varies with my location, and every 15-degrees latitude corresponds to the 1-sidereal hour (nb. in a continuous manner).
• A star is highest at its meridian crossing, in other words, when sidereal time is its RA.
  • At this time its ZA (zenith angle, i.e. degrees from zenith) equals its declination minus the latitude. (Nb. this follows from the fact that our zenith is equivalent to our latitude)…
  • …Furthermore: Since the altitude (i.e. local view) of the zenith is 90-degrees, then the altitude of a star at the meridian crossing equals 90-degrees minus ZA. However, see next point…
  • …Nb. ZA must be given an absolute form, so if the declination is greater than the latitude then the azimuth must be made to zero degrees, and otherwise to 180 degrees. In other words, in the former case the meridian crosses north of the zenith, and in the latter case, south.
  • To find the star earlier/later, rotate east/west by 15 degrees.

5: Where is the sun?

• The sun’s right ascension makes a full cycle (i.e. a sidereal day) over the course of a year.
• Because the Earth is moving along its orbit as well as rotating, therefore once it’s completed a complete rotation it must still turn a little further in order to be the same relative angle (i.e. the sun’s RA) to the sun.
  • Therefore a solar day, as timed by our clocks (aka local time), is longer than a sidereal day (thus 24 sidereal hours equals 23h-56m-4s).
• On September 21, sidereal time is roughly equal to local time (ignoring the conventions of time zones, etc). Obviously, after this date, sidereal time can be estimated by adding 4min per day (and vice versa too). Also, on the 21 of December/March/June sidereal time can be estimated by adding 6/12/18hrs (and obviously by next September, another whole day has been added, etc).

An example that makes use of this: When is Vega’s RA 18h 36m at midnight?

1. Since a star is highest at its meridian crossing, i.e. when its sidereal time is its RA…
2. …therefore we want Vega when sidereal time (ST) is 18:36. But we want this to occur at a local time (LT) of 24:00…
3. …therefore we want ST to equal LT + 18hr…
4. …which occurs on June 21. To get the extra 36m we calculate at 4m per day (i.e. the change between ST and LT) to get 9 days…
5. …therefore the answer is June 30.

[Via Analemma @ Wikipedia]: Although it can technically refer to the relationship between any two celestial bodies, it is generally used to mean that of the sun as seen from Earth. It can be drawn based on the position of the sun as seen from a fixed location at the same local time every day.

6: Tilt and seasons

• The Earth is 23.5 degrees tilted relative to the plane of orbit around the sun. Alternatively, the Earth is straight but moving around a tilted orbit.
• Obliquity (or axial tilt) is the angle between the equatorial plane and the orbital plane.
• It follows that the ecliptic (or sun’s movement around the celestial sphere) is at 23.5 degrees, and the two points when the ecliptic meets the celestial equator are called the vernal (March 21) and autumnal (September 21) equinox. The former by definition is the prime meridian, thus the sun is overhead at 0hr RA. The sun’s declination varies between these points to achieve a maximum declination of 23.5 degrees (i.e. south or north at the 12-month interval).
• E.g. after the vernal equinox, the ecliptic moves on the celestial sphere north away from the celestial equator, so that the latitude from which the sun is overhead moves north. During these 6-months, the declination of the sun is also the degrees of latitude away from the north pole which is always exposed to the sun.
• At equinox day the day/night is 12/12 hours everywhere.
• Tropics are limited by 23.5 degrees latitude north/south from the equator, which corresponds to the areas which see the sun reach at least one subsolar point (i.e. perceiving the sun to be at my zenith).

An example that makes use of this: How high is the sun at noon at (Athens) latitude 37.7 degrees north?

1. At the equinox: This is when the sun is 0 degrees declination everywhere, thus ZA (equals RA minus declination, or latitude minus declination) is 37.7 degrees, and altitude (equals 90 degrees minus ZA) is 52.3 degrees.
2. At the summer solstice: This is when the declination is at its maximum, 23.5 degrees, thus ZA equals 14.2 degrees, and altitude 75.8 degrees.
3. At the winter solstice: When the declination is at negative 23.5 degrees, thus ZA equals 61.2 degrees and altitude 28.8 degrees.

7: The Age of Aquarius

• The solar day is only 24h as a mean due to a few factors including:
  • Because the ecliptic is parallel to the equator near the solstices, therefore its eastwards motion is then the fastest.
  • Due to the elliptic shape of Earth’s orbit, it is slightly nearer and therefore moves slightly faster around January.
  • Precession, i.e. the wobble of the Earth’s axis in a westward direction. The north pole moves to the west with a radius of 23.5 degrees every 26,000 years relative to the stars.
  • Precession of the equinoxes follows from the precession causing the points of the equinoxes along the equator to move. From this it follows that the coordinates of the stars move, leading to a system of giving coordinates relative to epochs (generally the epoch J2000, i.e. on January 1 2000).
• Age of Aquarius refers to the fact that around 2600CE, the vernal equinox will enter that constellation (from the current Pisces).

8: The moon moves too

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