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Lab 2: Celestial Sphere: Background Notes

Sections

  1. Equatorial Coordinates
  2. Observing the Sky
  3. Horizontal Coordinates
  4. Paths on the Sky
  5. The Seasons
  6. Axial Precession

  1. Equatorial Coordinates

    1. The astronomical objects on the sky are all very remote and show no parallax to simple observations.

      Therefore, we an project all of them as viewed from the point-like Earth onto infinitely remote imaginary sphere which we call the celestial sphere.

      A globular map of the the celestial sphere is a celestial globe

      Astronomical objects on the celestial sphere can be located using equatorial coordinates.

      The animation below illustrates equatorial coordinates.

    2. Earth's equator is likewise projected from the center of the Earth to become the celestial equator and the Earth's axis is extended to become the celestial axis which its the celestial sphere at the north celestial pole (NCP) and the south celestial pole (SCP).

    3. Latitude and longitude are projected outward to make the equatorial coordinates, except for some changes.

      Everything on the sky can be located in equatorial coordinates.

      Extra-solar-system astro-bodies have approximatlely fixed equatorial coordinates at least over human lifetimes. There is slow variation due to axial precession (see below).

    4. In the equatorial coordinates, latitude becomes declination (dec) measured positive north and negative south. Dec is measured in degrees, arcminutes (1/60 of a degree), and arcseconds (1/60 of an arcsecond).

    5. Longitude becomes right ascension (RA) measured eastward from the vernal equinox in hours, minutes, and seconds---one hour being 15 degrees. Why hours,etc? Ask the Babylonians.

      Everyone find and touch celestial equator, NCP, SCP, and the vernal equinox.

      The celestial equator is a great circle---which is a circle that divides a sphere in half.

      The RA meridians are also great circles.

      What is the vernal equinox?

      The zero point of RA celestial globe as 0 RA or as the vernal equinox or as the point where the ecliptic crosses the celestial equator going north.

    6. What are ecliptic and vernal equinox? and

      In the inertial frame of the fixed stars, the Earth orbits the Sun.

      But in the non-inertial frame non-rotating geocentric frame of the Earth, The Sun orbits the Earth against the backgound of the celestial sphere on which all the extra-solar-system astro-bodies are approximately fixed.

      The ecliptic is the great circle path of the Sun on the celestial sphere.

      The Sun moves eastward and takes a year to complete an orbit.

      The vernal equinox is the point where the ecliptic crosses the celestial equator going north.

      When the Sun is at that point, it is the begining of spring---this point in time is also called the vernal equinox---vernal equinox has two meanings.

    7. Why is the ecliptic not on the celestial equator?

      Well the plane of the Earth's orbit---which is also the Sun's orbit around the Earth---is the ecliptic plane.

      The ecliptic plane cuts the celestial sphere at the ecliptic.

      The ecliptic pole is perpendicular to the ecliptic plane.

      The Earth's axis is tilted from the ecliptic pole by about 23.4 degrees---everyone knows the tilt, but few know what the tilt is respect to---it's to the ecliptic pole.

      Since the Earth's axis is titled from the ecliptic pole by 23.4 degrees, the Earth's equator and the celestial equator are tilted from the ecliptic by 23.4 degrees.

      The animation illuatrates the tilt of the ecliptic from the celestial equator.

  2. Observing the Sky

    1. The celestial globe cannot be to scale.

    2. The celestial sphere should be huge compared to the Solar System so that there are no angular shifts of stars (i.e., parallaxes) as one moves about in the Solar System. Solar System is a pinprick at the center of the celestial sphere.

    3. The Solar System itself is huge compared to the size of the Earth.

    4. The Earth is huge compared to us---we are microbes on a beach ball.

      This is much of the problem in understanding the sky.

      The ground to us is an infinite plane that cuts the celestial sphere in half---half, below and half above.

      Here we idealize the the ground as a plane that is exactly perpendicular to the vertical line to the Earth and that plane is tangent to the actual ground.

      As the Earth rotates around on its Earth's axis, the half we see keeps changing continuously---daylight and clouds also impede our view of the sky.

      The horizon is the great circle dividing sky in two.

      We see above not below it.

      The term horizon is also used for the dividing plane itself.

  3. Horizontal Coordinates

    1. Equatorial coordinates are great for cataloging the locations of astro-bodies since they are the same for all observers anywhere on Earth---remember Earth is a pinprick---and for significant length of time---they do have to be updated because of axial precession which we discuss below.

      With the proper instuments, equatorial coordinates are easy to use.

      But for just casual observation and telling people where something is when they are right there beside, equatorial coordinates are not so great---``where the heck is that RA and dec?"

    2. Horizontal coordinates (AKA local coordiantes) satisfy the need to locate astro-bodies quickly in a way easy to humans to understand and use.

      But a given set of Horizontal coordinates only locate an astro-body for one moment in time at one location.

      So they are no good for catalogs of location.

    3. First note that zenith is the point on the celestial sphere where the local vertical hits the celestial sphere---the zenith is the point that's right overhead.

    4. The horizontal coordinate altitude is the angle measured from the horizon on a great circle that goes from the horizon to zenith.

      Atitude can be approximately measured easily: a spread hand at arm's length subtends about 18 degrees.

    5. The horizontal coordinate azimuth azimuth is measured along the horizon east from due north.

      Find Polaris and right down along its altitude line is due north.

    6. The meridian (AKA the celestial meridian is a great circle that passes through due north, zenith, and due south.

  4. Paths on the Sky

    1. Herein we consider the paths on the sky of astro-bodies during a day from a geocentric-frame point of view.

    2. Every day the celestial sphere sweeps west and completes a period in about 1 day.

    3. The paths of the astro-bodies are generally small circles---circles that cut a sphere NOT in half.

    4. When an astro-body crosses the meridian it called transiting the meridian.

      The event of transiting is called a transit.

    5. But because the horizon cuts the celestial sphere in half, some astro-body small circle paths are cut in half---and some are not.

      What decides?

      First, to be simple, let's just consider a Northern Hemisphere case: the Southern Hemisphere is sort of the mirror image case.

      Next note that NCP is one thing on the sky with a fixed altitude with respect to time (but not location on Earth) since it does NOT rotate with the celestial sphere.

      A little geometry shows that NCP's altitude is equal to the latitude of the observation location.

        astlec/lec002/ncp_altitude.png: Internet Explorer fails.

        NCP altitude and latitude.

      If a small circle is too nearer the NCP than the NCP's altitude, then small circle never falls below the horizon and its astro-body is always above the horizon.

      If a small circle is too nearer the SCP than an angle equal to the NCP's altitude, then small circle never rises above the horizon and its astro-body is always below the horizon.

      Astro-bodies that never rise or set are called circumpolar objects: they could be circumpolar stars circumpolar constellations.

      So any astro-body closer to the NCP than an angle equal to the local latitude is circumpolar and never sets.

      And any astro-body closer to the SCP than an angle equal to the local latitude is circumpolar and never rises.

      All other astro-bodes are NOT circumpolar and rise and set every day on the horizon.

      These non-circumpolar astro-bodes follow arcs on the sky which are the visible part of their small circle paths.

      Extra-solar-system astro-bodes always rise and set at the same azimuths (which are equal in magnitude and opposite in sign) at least approximately over the course of a human lifetime.

      Solar-system astro-bodes move continuously on the celestial sphere and this is noticeable over very short time periods.

    6. How does the Sun move on the sky.

      Well everyday it circles west on a small circle. So it follows an arc on the sky when above the horizon.

      But it also moves relative the celestial sphere thought of as attached to the fixed stars on eastward the ecliptic.

      It takes one year (approximately 365.25 days) to complete it's path on the ecliptic.

      So it moves at about 1 degree per day---the ancient Babylonians may have chosen to divide the circle into 360 degrees to get this angular velocity.

      This means the small circle path of the Sun varies north and south of the celestial equator by the Earth's axial tilt of 23.4 degrees.

      The maximum declination is called the summer solstice.

      Minimum is called the winter solstice.

      When the Sun is on the celestial equator, it is called an equinox: when the Sun is going north it is the vernal equinox and when it is going south is the fall equinox.

      The solstices and equinox happen about 91 days apart.

      Fiducial dates are Mar21 vernal equinox, Jun21 summer solstice, Sep21 fall equinox, and Dec21 winter solstice.

      The actual dates of the solstices and equinoxes vary a bit for several reasons.

      The main one is that the solar year (vernal equinox to vernal equinox) is NOT a whole number of days: it is about 365.242 days---it varies a bit due all kinds of things.

      But calendar years have whole numbers of days.

      We fixed things up to keep solar year count and calendar year count synchronized over long periods of time by alternating the length of the calendar year between 365 days (ordinary years) or 366 days (leap years) in a way prescribed by the Gregorian calendar.

      Because of the short-term desynchronization of the solar year count and calendar year count the solstices and equinoxes can move a calendar day or so off the fiducial dates.

      The Solstices and equinoxes mark the beginnings of the astronomical seasons that their names indicate.

      Because the Sun moves continuously eastward relative to the fixed stars, one can also say that the fixed stars move continuously westward relative to the Sun

      This means that the solar times for fixed stars happen earlier every day: rises times, setting times, and transit times.

      For example, say a particular star transits the meridian at the same time as the Sun (i.e., at solar noon) on a given day.

      This means the star and Sun are at the same RA.

      Next day, the star has moved westward from the Sun and will transit the meridian before solar noon.

      Of course, earlier and earlier times eventually cycle back to the times where they started.

      If a star rises with the Sun on a given day (called its heliacal rising), then it will rise earlier and earlier until a year later is it is rising with the Sun again.

    7. As the Sun moves eastward on the ecliptic it passes though various constellations.

      Which constellations the Sun and ecliptic passes through depends on how you define constellation and how you define passing through a constellation.

      But in traditional astrology, their are 12 constellations on the ecliptic which are called the zodiac constellations: Aries, Taurus, etc.

      On average, it takes about 30 days for the Sun to move through a zodiac constellation.

      You cannot observe the zodiac constellation the Sun is in, except you can see part of it at sunrise and sunset.

  5. The Seasons

    1. The astronomical seasons are just marked by Sun's location on the ecliptic.

    2. But these seasons are also the climatic seasons.

      Why?

    3. Let's talk energy flux: energy per unit area per unit time.

      The higher the energy flux from the Sun on a give location, the higher the temperature tends to be.

      Only "tends" because many other factors affect temperature.

    4. On a surface perpendicular to the a href="http://en.wikipedia.org/wiki/Sun">Sun's beams the energy flux absorbed by the surface is is highest.

      As the surface is tilted away from the perpendicular, the energy flux absorbed by the surface decreases since the total energy captured by the surface decreases.

      The total energy captured goes to zero when the zero is parallel to the beam.

      Everyday experience confirms this.

      The Sun is more heating than the Sun at sunrise or sunset.

    5. The Sun beams most directly on the Northern Hemisphere in daytime near the summer solstice when the northern end of Earth's axis is tilted toward the Sun.

      The Sun beams most obliquely on the Northern Hemisphere in daytime near the winter solstice when the northern end of Earth's axis is tilted away from the Sun.

      Thus, Northern Hemisphere summer is hotter than Northern Hemisphere winter.

      At the equinoxes, the Earth's axis is perpendicular to the Sun-Earth line and the Sun beams down most directly on the equator.

      Then we have intermediate heating in the Northern Hemisphere.

      Thus the axial tilt of the Earth explains the main facts of the climatic seasons.

      The Southern Hemisphere seasons are explained the same way by a mirror argument.

    6. But now you say, the solstices and equinoxes mark the beginnings of climate seasons not their midpoints as the tilt argument suggests.

      Well it takes time to accumulate and lose heat energy.

      This results in a lag time of about a month between time of a particular energy flux from the Sun and the corresponding temperature.

      The lag time is called seasonal lag.

      The result of the seasonal lag is that the solstices and equinoxes do mark the beginnings of climate seasons on average as we usually think of them.

    7. But what of the Sun-Earth distance? It varies. Does that not effect the seasons.

      The variation is not large. Only about 3.4 %.

      It must have some small effect on climate, but much less than the axial tilt.

      In fact, the Earth is closes to the Sun in January and farthest in July.

      Clearly, the effect of the axial tilt overrides the distance effect totally in the Northern Hemisphere.

  6. Axial Precession

    1. The Earth's axis precesses relative to the inertial frame of the fixed stars with a period of about 26,000 years---the period probably varies a bit.

    2. Precession means the Earth's axis sweeps out a double cone with apex at the Earth center.

    3. The cone symmetry axis is the ecliptic axis.

      The angle from the ecliptic axis varies a bit for many reasons, but stays about 23 % over time.

    4. Why, O why is there axial precession?

      The short answer is that the Earth is oblate: it bulges at the equator (see Wikipedia: Earth's rotational bulge).

      The gravitational forces of the Sun and Moon and to a much lesser extent the planets tug unequally on the bulges on opposite sides of the Earth (i.e., they exert a tidal force on the Earth's rotational bulge) and this results in precession because the Earth is rotating.

      The effect cannot really be made intuitively obvious.

      But it's not uncommon.

      A spinning toy top precesses for similar reasons.

      A spun coin rings down precessing for somewhat similar reasons---but this is is very actually tricky case.

      Gyroscopes exhibit precession as one of their main features.

      Rotationaly systems with unfixed axes are just hard to understand even though they are not uncommon.

    5. The axial precession complicates the equatorial coordinates.

      They are fixed to the Earth's axis and equator.

      But that means that all the astro-body equatorial coordinates will continuously change with time just due to the axial precession aside from other factors.

      So one has to update one's catalog of locations in equatorial coordinates to allow for axial precession.

      In the old days, this was a major chore and done only every few decades.

      Nowadays, computers can do it pretty quickly.

      Nevertheless, whenever you make a precise location observation you have to report not just equatorial coordinates but also their epoch.

      TheSky gives year 2000 equatorial coordinates and current equatorial coordinates in its information dialog boxes for items.