Lab 2: Celestial Sphere / Key

The answers are intended to be super-perfect. They may not actually be that.

Full marks can be given for anything that looks reasonably complete.

  1. Question 1:

    During a solar day, the Earth-Sun line shifts eastward by about 1 degree due to the Earth's rotation around the Sun. This means that the Earth must rotate more than 360 degrees relative to the fixed stars to bring any point on Earth back to solar noon. So the solar day is longer than the rotational period relative to the fixed stars (i.e., the sidereal day).

    To be quantitative: 

           R_sid*t=R_orb*t+360

            where t is the solar day.

       And so

       t=360/(R_sid-R_orb)=t_sid/(1-t_sid/t_orb)=86400.0 s

         where t_sid = 86164.1 s 
         and t_orb = 365.256363004*86400 s = 31558149.763545599 &cong π*10**7 s.

      Fortran code:

      print*
      tsid=86164.1d0  ! Sidereal day
      torb=365.256363004d0*86400.d0
      tsol=tsid/(1.d0-tsid/torb)
      print*,'torb,tsol'
      print*,torb,tsol
!  31558149.763545599        86400.000340327417

    So we get the right answer.

    We can also see that the relative difference between the two days is of order t_sid/t_orb ≅ 1/365.

    The diagram you will have to imagine.

  2. Question 2: A great circle divides a sphere into equal parts and a small circle divides into unequal parts.

  3. Question 3: great circle.

  4. Question 4: a) never rise or set / never cross the horizon during the rotation of the celestial sphere. b) depend on your location on Earth.

  5. Question 5: 0 hours, 0 degrees, 90 degrees, -90 degrees.

  6. Question 6: about 80 degrees, 0 degrees, 90 degrees, -30 degrees.

  7. Question 7: β HYI, -77 deg, 20'.

  8. Question 10: Eastward by about 1 degree per day or more precisely 360/365.25 degrees per day. Those good old ancient Babylonian astronomers may well have chosen 360 degrees in a cirlce precisely to get make the angular velocity of the Sun on the ecliptic nearly a nice round number. The Sun's motion on the celestial sphere was fundamental to their astrology (which is nearly our astrology). They didn't choose the degree size to give precisely 1 degree per day since that would make an inconvenient number of degrees in a circle for calculation. Of course, we don't know why they chose 360 degrees in a circle---they never tell us anything.

  9. Question 11: Check out Wikipedia: Zodiac: Table of Dates shows IAU constellation boundaries.

  10. Question 12: ealier. I like to say "earlier and earlier" but does that make any sense?

    eastward by about 1 degree per day.

  11. Question d:
    1. An horizontal coordinates of astronomical object at one place on Earth and time apply only to that time and place. So catalogs of values would be good for only one place and time in general. They would be of little use to anyone.

    2. When you want to locate something in the sky right here, right now, horizontal coordinates to that.

      You can easily estimate them to tell someone else where for look for example for astronomical object.

      Nowadays, there is lots of software will instantly give you the horizontal coordinates that it computes from your time and location, and the stored equatorial coordinates.

    3. Question 14:

      1. on the celestial equator.
      2. due east and due west. 12 hour each ideally.
      3. north of east and west. Day.
      4. south of east and west. Night.
      5. at the north or south pole.
      6. on the equator.
    4. Question 15:

      1. Alt=Lat=36'10.5''=36.175 degrees for Las Vegas, Nevada.
      2. An astronomical object has to be closer in angle to the NCP than the NCP's altitude value Alt since otherwise it would rise and set. To be closer to the NCP than Alt implies that Dec ≥ 90-Alt=90-Lat.
      3. For Las Vegas, Nevada, Dec ≥ 90-Lat ≅ 54 degrees. From the all-sky sky map, we find the on-average circumpolar constellations are Draco, Cameloparalis, Cassiopeia, Cepheus, Ursa Major, and Ursa Minor
    5. Question 16:

      1. Well the NCP to celestial equator axis is 90 degrees. Add latitude and subtract declination to get Alt_N=90+Lat-δ

      2. From due south, the formula is Alt_S=180-Alt_N=90-Lat+δ.

      3. Alt_S_SCP = -Lat.

      4. Lat = -Alt_S_SCP = -23.4 degrees or 23.4 degrees south latitude and you are on the Tropic of Capricorn.

    6. Question i: 1) πR**2, 2) 2πR**2, 3) I/2.

    7. Question 18:

      These question are hard to answer concisely. It actually takes working them over and over again to get really good answers. It's good for the students to try. One can mark generously.

        The average insolution of the northern and southern hemisphere varies periodically with the periodically varying tilt of the Earth's axis with respect to the Sun. When the north end of the axis is tilted toward from the Sun, the insolution is higher in the north and its summer there and lower in the south and its winter there. When the north end of the axis is tilted away, the reverse situation holds. Near the equinoxes when the tilt is relatively close to 90 %, the insolation is about equal in north and south, and we have the intermediate seasons of spring and fall. The period of oscillation of the tilt is 1 year and that is why the cycle of the seasons is 1 year. The cycle in the south is 180 degrees out of phase with the cycle in the north.

        Obviously this question can be answered in many different ways. Since it is supposed to be the short answer, it can't be complete---without remarkable compression.

      1. Well it must have some effect since the insolation at the top of the atmosphere varies with distance from the Sun. However, the eccentricity seems to be a rather minor effect since perihelion occurs in the northern hemisphere winter and it doesn't stop winter there and aphelion occurs in the northern hemisphere summer and it doesn't stop summer there. The eccentricity must act to moderate the northern hemisphere seasons and exagerate the southern hemisphere seasons. But there are so many other secondary effects on the seasons that is hard to say how important the eccentricity is. Clearly if the eccentricity were much larger it would have a much larger effect.

      2. Diagrammatically, the situation is clear. In the summer, the North Pole is tilted toward the Sun and the Sun just hangs in the sky at constant altitude moving on a circle parallel to the horizon. A bit off the north, the sun moves on circle that is titled with respect to the horizon, but never sets. In the winter, the North Pole is tilted away from the Sun and the Sun never rises above the horizon for locations near the north pole. The situation at the South Pole is the mirror image.

    8. Question k: vernal equinox, summer solstice, fall equinox, winter solstice.

    9. Question l:

      Las Vegas, Nevada 36 deg, 10.5' = 36.175 deg. 

          ---------------------------------------------------------------------------------------------------------------
    Table of Solstice and Equinox Data
    ---------------------------------------------------------------------------------------------------------------
    Sun Position      Fiducial Date     RA          DEC          Altitude of the Sun in Las Vegas
                                       (hr)      (degrees)                 Solar Noon
    ---------------------------------------------------------------------------------------------------------------
    vernal equinox    Mar21              0           0                         53.8
    summer solstice   Jun21              6          23.4                       77.2
    fall equinox      Sep21             12           0                         53.8
    winter solstice   Dec21             18         -23.4                       30.4
    ---------------------------------------------------------------------------------------------------------------

    10. Question m: 

            print*
      p=25771.0d0  ! Julian years it seems http://en.wikipedia.org/wiki/Axial_precession#Values
      rate1=360.d0/(p*365.25)
      yearsid=365.256363004d0  ! http://en.wikipedia.org/wiki/Year#Sidereal.2C_tropical.2C_and_anomalistic_years
      yearsol=365.24219d0
      rate2=360.d0/yearsid
      yearsolc=360.d0/(rate1+rate2)
      dela=rate1*yearsolc
      print*,'rate1,rate2,yearsolc,yearsol,dela'
      print*,rate1,rate2,yearsolc,yearsol,dela
!  3.82455583162297325E-005  0.98560911311504673        365.24219015261588        365.24218999999999       1.39688914830293394E-002

      The period is called the solar or tropical year.

    11. Question n:

      The equatorial coordinate system is tied to the direction of the Earth's axis. This makes it very convenient for locating objects from the Earth as it rotates. But it means that the equatorial coordinates of all astronomical objects beyond the Solar System change as time passes since the whole equatorial coordinate system grid of equatorial coordinates shifts with the axial precession of the Earth and also the change in the axial tilt angle of the Earth.

      Another effect the Sun and astronomical objects beyond the Solar System have physical motions through space. For remote objects, these motions have negligible effect on their equatorial coordinates for very long periods. However, for nearer objects the effect is not negligible and the equatorial coordinates of these objects must be updated for that reason.