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ASTR 3130, Majewski [SPRING 2015]. Lecture Notes

ASTR 3130 (Majewski) Lecture Notes


REFERENCE: Kaler Chapter 3.


Recall some important concepts:

  • A sidereal day is about 4 minutes shorter than a solar day.

  • Thus, in one solar day, the Earth has actually rotated with respect to the stars by 361o.

  • Thus, at the same solar time each day, the sky advances 4 minutes of hour angle compared to the previous day. Thus, the same stars are progressively moving about 4 minutes westward compared to the Sun per day.

  • Thus, the Sun is continually moving eastward with respect to the stars.

  • 4 minutes per day is about two hours per month and is 24 hours per year, after which the cycle repeats.

  • It thus follows that the Sun's right ascension is continuously changing throughout the year.
The same holds true for the Sun's declination -- it changes continuously throughout the year. This is a consequence of the Earth's tilt -- the difference in its rotational and revolutionary axes.

Recall that the plane of the Earth's orbit determines the apparent path of the Sun among the stars:

  • The plane of the Earth's orbit is the ecliptic plane.

  • The projection of the apparent path of the Sun on the celestial sphere is called the ecliptic.

  • The difference between the ecliptic and the celestial equator is determined by the difference in the rotational and revolutionary axes of the Earth, and is called the obliquity of the ecliptic .

    • The value of the obliquity of the ecliptic is 23.5o.

  • Note that most of the other planets, most of the moons of these planets (including the Earth's moon), and the rings of Saturn are also found to like in the ecliptic plane.

  • The zodiac is a band of constellations lying along the ecliptic. There are 13 constellations that the ecliptic crosses, but only twelve are traditionally recognized (Ophiucus has been left out).

    These twelve signs represent astrological "houses", each about a month long, through which different solar system objects can pass (note that a fourteenth constellation, Cetus, lies close to the ecliptic, so sometimes planets can be found in Cetus as well).

  • The Sun's position in these different houses at different times of the year define the traditional "months" assigned by astrologers to different zodiacal signs.

    • But note that this assignment is from several thousand years ago. Due to precession of the direction of the Earth's tilt in the sky, there is a slight yearly shift in the timing of the Sun's appearance at any given position along the ecliptic.

    • The precession has a net shift of 50.3 arcseconds per year and makes one complete cycle in about 26,000 years.

    • That the "assigned" months for different zodiacal signs have shifted from their original positions is one way to question those who ascribe astrological significance to the zodiac...

  • You can see from the above figure that the view of the night sky must change as Earth moves in its orbit around the Sun.

    • Thus we see why many constellations are seasonal.

    (left) A typical summer sky in the United States looking towards the Southern horizon. (right) A typical winter sky in the U.S. facing the same direction.


    • Even for profesional astronomers, it is useful to have some basic knowledge of bright stars and key constellations.

    • Most of you probably know the circumpolar constellations Cassiopeia, Ursa Major, Ursa Minor and Cepheus, and the star Polaris.

    • Also good to know is the Summer Triangle (Deneb, Vega, Altair) and the associacted constellations (Cygnus, Lyra, Aquila).

      Left image from Right image frim
    • And the Winter Hexagon and associated constellations.

      From .

    Recall that the position of the celestial equator on the sky at any place remains constant.

    • At the meridian, it is 90o-Latitude above southern horizon.

    Therefore, since the ecliptic is tilted with respect to the celestial equator, the ecliptic has a varying height above the southern horizon over the course of the day.

    • This means that solar system objects that lie along the ecliptic will have different positions relative to the equator and the southern horizon at different times of the year.

    The place where the ecliptic and celestial equator cross at the time when the Sun is moving northward with respect to the equator is called the vernal equinox.

    • The point of the vernal equinox is in the constellation Pisces.

    • This point is also used to define the location of 0 hours of right ascension, which then increase eastward along the celestial equator.

    • Note that the Sun reaches the point of the vernal equinox on or near March 21. We call that day by the same name, vernal equinox (the word "vernal" is Latin for "spring"), although the expression spring equinox is also used.

    • On the opposite side of the sky, we have the point where the ecliptic and equator cross again, but when the Sun reaches this point it is moving south. This point on the celestial sphere (and the day on which the Sun reaches it -- on or near September 21) is called the autumnal equinox.

      • THOUGHT PROBLEM: Why is the terminology "spring equinox" and "autumnal equinox" for those specific dates of the year "north-centric"?

    • The picture below shows an example of the relative position of the Sun, Venus, celestial equator, ecliptic and lines of right ascension.

      • THOUGHT PROBLEM: Approximately what day of the year is represented in the figure?


The 23.5o tip of the Earth's equator with respect to the plane of the Earth's orbit is the cause of the Earth's seasons.

In (a) we see the apparent path of the Sun on the celestial sphere. In (b) we see the actual relation to Earth's rotation and revolution.

It is also the reason why we have different amounts of sunlight on different days of the year.

The cartoon attempts to summarize how the daylight and seasons are connected:

Some important points:
  • Sep. 23 in the figure is near the autumnal equinox, while March 21 is near the vernal equinox.

    • On these days, the Sun is on the celestial equator and (almost) everyone on Earth has about 12 hours with the Sun above and below horizon.

      • THOUGHT PROBLEM: Where are the exceptions to the above statement and why?

    • Note that "equinox" is Latin for "equal night" (to day).

    • Note that the rays of the Sun, which is located on the celestial equator, strike the ground exactly vertically for observers on the equator on this day, and the Sun is at the zenith at noon:

  • On or around June 22, we have the summer solstice:

    • The direction to the Sun is 23.5o above the celestial equator.

    • Here the days are longer than the nights for the Northern Hemisphere and the nights are longer than the days for the Southern Hemisphere. The longest day in the Northern Hemisphere is when the Sun reaches +23.5o.

    • Not only are the days longer for northerners, but the rays of the Sun strike the ground more nearly vertically, meaning they are less spread out (i.e., more concentrated), and thus more energy (e.g., heat) is received per square foot.

    • On or about June 22, the Sun's is at the zenith at noon for observers at latitude +23.5o. This latitude is called the Tropic of Cancer.

    • For observers above latitude 90o-23.5o=66.5o, there is continuous sunshine. This latitude is called the Arctic Circle.

    • For observers below -66.5o latitude, there is continuous 24 hour darkness. This latitude is called the Antarctic Circle.

  • On or around Dec. 22, we have the winter solstice:

      • The direction to the Sun is 23.5o below the celestial equator.

      • Here the days are longer than the nights for the Southern Hemisphere and the nights are longer than the days for the Northern Hemisphere.

      • Now the Southern Hemisphere enjoys longer nights and greater concentrations of energy while the Northern Hemisphere is in winter.

      • This is the longest day for the Southern Hemisphere and the longest night for the Northern Hemisphere.

      • On or about Dec 22, the Sun's is at the zenith at noon for observers at latitude -23.5o. This latitude is called the Tropic of Capricorn .

      • For northerners, the rays of the Sun reach the ground rather obliquely . With the light spread out more, we actually receive less per square foot, and thus we are less warm.

      • For observers above the Arctic Circle (latitude 90o-23.5o=66.5o) there is continuous, 24 hour darkness, and for those below the Antarctic Circle there is 24 hour daylight.

    • Keep in mind the special latitudes of the Earth, which are defined astronomically.

The number of hours of daylight and the angle at which sunlight strikes Earth's surface determine the "insolation", or the amount of sunlight incident on a unit area of the Earth' s surface during 24 hours. In the end, this differential heating is responsible for the occurrence of seasons in Nature:
  • At this website you can see a summary of the seasonal year, the position of the Sun, and how sunlight is projected onto the ground for a Northern observer.

  • The change in the geographic shadow distribution caused by the Earth's tilt is quite dramatic (even though the shadow always covers exactly one hemisphere of the Earth).

    • Here are two images of the way the shadow is distributed at about 2 PM Eastern time on August 1 (left) and December 1 (right). The Earth's surface moves eastward through the shadow. You can immediately tell from the image which latitudes are receiving more sunlight in a 24 hour period. Click on the thumbnails for an expanded view.

    • At this website you can see what the shadow of the Earth looks like right now (or any other time).

NOTE: A common misperception is that the Earth's seasons are caused by the distance of the Sun from the Earth.

  • If this were true, the Northern and the Southern Hemispheres would have simultaneous seasons. Instead, they are six months apart.

  • The Earth is actually closest to the Sun in January!


  • Use geometry to prove that the Arctic Circle should be at +66.5.o.


With knowledge we have already discussed, you should now be able to estimate the celestial coordinates of the Sun for different days of the year:
  • On March 21 and Sep. 22 the direction of the Sun in the sky is the same as the celestial equator.

      --> The Sun is at declination = 0o on the autumnal and vernal equinoctes.

  • On Dec. 22, the direction of the Sun is 23.5o below the celestial equator (Sun transits overhead at the Tropic of Capricorn).

      --> The Sun must have a declination of -23.5o on the winter solstice.

  • On June 22, the direction of the Sun is 23.5o above the celestial equator (Sun transits overhead at the Tropic of Cancer).

      --> The Sun must have a declination of +23.5o on the summer solstice.

  • Astronomers have defined the line of 0 hours of right ascension as that line of right ascension corresponding to the point of the vernal equinox on the celestial sphere.

    • From there, use knowledge that the Sun moves Eastward (i.e. later) on the celestial sphere by about 4 minutes per day to ascertain the Sun's Right Ascension on any day.

    • Note that the RA of the Sun is the same thing as saying what RA is transiting at noon...

  • As we have learned, the height of an object at transit, where the object rises and sets, and the number of hours that object is above the horizon are all determined by the declination of the object. Thus, the -23.5o to +23.5o range of declination for the Sun affects these various aspects of the Sun's diurnal path through the sky.

    The Sun's daily path for latitude +40o at monthly intervals.

    This movie demonstrates the seasonal variation in the Sun's daily path.

  • Astronomers are frequently more interested in knowing what is happening at night. For example, what hours of right ascension will be up at night.

    • The above RA/DEC coordinate map of the Sun is useful for determining what is happening 12 hours later, e.g., what RA is transiting at midnight.

    • You can also get a sense of what is happening to the ecliptic at midnight by thinking about what is happening 12 hours after the Sun.

      • In summer the Sun transits above the celestial equator, so planets transiting near midnight will transit below the equator and will be low in the night sky.

      • In winter the Sun transits below the celestial equator, so planets transiting near midnight will transit above the equator and be high in the night sky.

Sunrise and Sunset

REFERENCE: Kaler Chapter 7, 13.15-13.25.


When the is below the horizon, the atmosphere of the earth can still catch rays of the sun and scatter them to visibility by an observer. This is called twilight.

There are three types of twilight that have been defined:

  • Civil Twilight: begins and ends when sun is 6o below the horizon.

      Used in courts of law, for example.

  • Nautical Twilight: begins and ends when sun 12o below the horizon.

      Brightest stars as well as the sea horizon are visible.

  • Astronomical Twilight: begins and ends when sun 18o.

      6th magnitude stars visible at the zenith.

Obviously, the duration of twilight depends on how quickly the Sun reaches the various angles below the horizon, and so depends on the angle that the Sun is moving with respect to the Sun.

  • Thus, the length of twilight depends on the day of the year and the latitude of the Earth.

  • For example, near the equinox, an equatorial observer has the shortest possible twilights because the Sun is moving perpendicular to the horizon. However, around days polar observers witness twilights that last for days.

Tables of twilight times throughout the year may be found in Norton's 2000.0 (pp. 49-52).

Tables of sunrise/sunset times throughout the year may also be found in Norton's 2000.0 (pp. 45-48). When using these tables, remember to keep track of daylight savings time.


There are a number of atmospheric phenomena that come into play when objects rise and set. Though it happens to all celestial objects, the effects are most commonly seen with the Sun.

As objects reach the horizon, the airmass grows substantially.

  • Recall from previous lecture that the airmass is 1 at the zenith and reaches 38 at the horizon.

  • The substantial increase (38 times) in the amount of atmosphere in front of a celestial object means that there is substantially poorer seeing for objects near the horizon.

    • Professional astronomers rarely work when objects are above 2 airmasses.

    • THOUGHT PROBLEM: What zenith distance corresponds to 2 airmasses?

  • There is substantially more scattering of light at all wavelengths.

    • This means objects are always fainter when they are near the horizon and brighter near the zenith.

      • You know from experience that the Sun can be viewed directly at sunrise/set but not when the Sun is overhead.

    • Blue light is always scattered more than red light, thus, objects become redder in color as they approach the horizon, because less red light is lost than blue light.

      • You know from experience that the Sun and Moon often look very red at sunrise/set.

  • Because the atmosphere is a transparent medium of increasing density as you approach the Earth, the atmosphere must also refract light according to Snell's Law, just like a lens.

    • Atmospheric refraction has the effect of making every object appear higher in the sky than it really is.

    • The effect is more pronounced as you look to lower altitudes (higher airmasses) because of the greater pathlength the light travels through the refracting medium.

    • A table of the amount of atmospheric refraction as a function of altitude is given on page 43 of Norton's 2000.0.

      Recall that the angular size of the Moon and Sun are about 30 arcmin. This means that when the Sun and Moon look like they are on the horizon, they are in fact already below the horizon!

      Thus, refraction lets you see things when they are already below the horizon (by up to 34.2 arcminutes).

        Note that this discussion applies to the true horizon, as seen from sea or towards flat horizon.

    • A number of other effects result from differential refraction:

      • Because the amount of refraction increases as objects approach the horizon, it has the effect of making objects appear to slow down as they approach the horizon.

        • From experience you know that sunsets last longer than the 2 minutes they should take to go from the time the Sun first touches the horizon to fully being below.

        • THOUGHT PROBLEM: Where does the value 2 minutes come from in that previous statement?

      • Differential refraction make the Sun and Moon appear to flatten as they reach the horizon. The image below explains why.

        The upper limb of the Sun is refracted less than the lower limb of the Sun. Thus, the lower limb is refracted by amount B which is larger than the amount A. The difference can be as much as one fifth the true angular diameter of the Sun.

        Flattening of the image of the Sun. From .

      • Refraction affects blue light more than red light (note physical difference between scattering and refracting).

        • Thus, blue light is always refracted to higher degrees than green light, and we have separation of "images" of the setting source sorted by wavelength.

          For example, stars will look like little spectra.

          This is an image taken of the star Polaris from a latitude of 52 deg ( high in the sky is Polaris when this picture was taken??). Differential atmospheric dispersion is spreading out the image of the star vertical to the horizon -- dispersing the light like a prism. It is because of this effect that it is difficult to obtain good images of astronomical objects near the horizon, unless one uses an Atmospheric Dispersion Compensator (ADC), a prismatic device that counteracts the atmospheric effect. From .

          For this reason, it is very important to understand how to orient a slit for slit spectroscopy if you want to capture all of the light from a source; the ideal orientation of the slit is vertical to the horizon.

        • The above image is for a broad wavelength region, but even for imaging using filters, if they are broad enough in wavelength, will still have the images spread out because of the differential refraction from the blue side to the red side of the filter.

          The amount of vertical smearing of images taken in red, green and blue broadband filters (i.e., filters some 900 Angstroms wide). Note that the problem is worst in the blue. The black curve is for a 2700 Angstrom wide filter covering the visible part of the spectrum (i.e., red, green and blue). From .
        • For extended sources, things are of course, complicated.

          Schematic illusgtration of differential refraction of Saturn, which, if uncorrected, will give a vertically color-sorted smear, not the clean image of the planet desired. From .

          When the Sun is near the horizon, it will look like a series of different Suns, as shown.

          Setting sun from Torrey Pines, CA by Andrew Young.

        • This can result in a phenomenon known as the green flash, as blue/green light is refracted into a blue/green image of the Sun that is higher than the orange/red image of the Sun.

          As the Sun sets, the orange/red Sun sets before the blue/green Sun, and, for a brief time, one can often see the upper limb of the green solar image "flash" as the last bit of Sun to sink below the horizon.

          Best seen over the ocean, lasts a few seconds, although can last much longer (up to 30 minutes or so) near the poles of the Earth (WHEN??).

        On May 19, 1999 Pekka Parvianen photographed this green flash from Finland.

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    Line drawings copyright © 1974,5 by Edmund Scientific Corp. Zodiac and axis tilt drawings copyright © by Nick Strobel. Pictures of the eye either copyright © 1997 Ed Scott and © 2000, 2001, or © The IESNA Lighting Handbook, 9th Edition. Star locating drawings copyright © 1974,5 by Edmund Scientific Corp. Green flash photo copyright © 1999 Virtual Finland. Differential refraction photo Andrew T. Young ( All other material copyright © 2002,2008,2012,2015 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 313 and Astronomy 3130 at the University of Virginia.