Laboratory Exercise #2 – The Celestial Sphere

 

Purpose:      Learn to use a geocentric model of the celestial sphere for purposes of celestial navigation and sky observing.  Given any location, date and time on the Earth, find the constellations and stars in the heavens.  Given the constellations and stars in the sky at a specific time and date, find the geographic location.

 

Introduction

The celestial sphere is one type of conceptual model.  It models the universe from the point of view that the Earth is the center of the universe.  Some ancient astronomers imagined that all the bodies in the universe including the Sun, Moon, planets and stars, are attached to a giant sphere, with the Earth at its center.  The stars were thought to be fixed to the sphere while the Sun, Moon, and planets moved along its inner surface.  This seemed to be intuitive based upon the simplest observations on any clear night.  This model of the universe is not a valid model; however, it is still useful for certain applications.

 

The geocentric celestial sphere is a handy way of determining the location of various celestial objects.  If you look at the appropriate place in the sky, as indicated on the celestial sphere, the desired object appears in your field of view, with or without a telescope.

 

Until the development of the atomic clock in the late 1960’s, monitoring the positions of the stars on the celestial sphere was an accurate means of timekeeping.  The study of the motions of the planets on the celestial sphere was fundamental in the development of Newton’s theory of gravity.  The observation of small deviations from the expected positions of some of the planets on the celestial sphere led to the discovery of Neptune.  Knowledge of the celestial sphere also permits one's position on the Earth to be precisely determined, something which is of particular importance in navigation.

 

Getting familiar with Farquhar Globe:

The Farquhar globe is the name given to the globe-within-a-globe mechanical model of the celestial sphere.  The outer globe represents the celestial sphere and the small, inner globe represents Earth.  Inside the larger celestial globe is a small, yellow ball attached to the end of a long, curved rod.  This ball represents the Sun. If you turn the knob to which the rod is attached you can simulate the motion of the Sun through the sky over the course of the year.  The path that the Sun traces in the sky is known as the ecliptic.

 

The metal ring on the stand that circles the center of the celestial sphere is the horizon ring. Look through the transparent celestial globe, past the Earth-globe to the far side of the celestial globe, to view the sky as it would be seen from Earth.  The Earth-globe is mounted on an axial rod which is connected to a knob at the bottom of the globe—this is the Earth-knob—which can be turned to rotate the Earth-globe.  The rod represents the axis on which the Earth rotates.  (The point at which the Earth's axis of rotation connects to the bottom of the celestial sphere is called the south celestial pole.  At the opposite end would be the north celestial pole.)  The Earth-knob should only be turned in a clockwise direction (as viewed from outside the globe), as if you were tightening a screw.  This is the direction in which the Earth actually rotates.  (Rotating the Earth-globe in the opposite direction may disassemble the globe.)  Rotating the Earth by means of the knob is equivalent to holding the knob and, thus, the Earth stationary, and rotating the celestial sphere.  Thus, we can consider the daily rotation of the Earth and the apparent rotation of the stars around the Earth as equivalent motions. 

 

Let’s try to locate some stars on the celestial sphere.

 

Procedure

Use the celestial sphere to answer the following questions.  Note that all questions or problems are repeated on the answer sheets given at the end of this laboratory exercise.  Separate the answer sheet pages from this write-up and write your answers on them.  This is what should be handed into your instructor.

 

Please note that the instructions below are for the larger celestial spheres.  Unfortunately, there are not many of these left, and the university is only supplying the smaller celestial spheres because of the costs associated with them.

 

1.) Stars on the sphere are represented by small circles of various sizes.  The larger the circle, the brighter the star.  What is the name of the brightest star in the constella­tion of Cygnus?  What is the name of the brightest star in the constella­tion of Lyra?

 

2.) Other objects such as galaxies and globular clusters are also marked on the sphere.  Which globular cluster is located near the constellation of Hercules?

 

The knob near, but not at, the north celestial pole controls the motion of the sun.  This is the sun-pointer knob.  Using the appropriate knobs on the sphere, rotate the Earth and move the Sun around the celestial sphere.  Notice that the Sun moves along a well-defined line on the celestial sphere.  This line is called the ecliptic.  The ecliptic corresponds to the plane defined by the Earth's orbit around the Sun.  We know, of course, that the Earth orbits around the Sun, but as viewed from Earth, the situation appears reversed: the Sun appears to orbit around the Earth.  (When considering the celestial sphere, it is more convenient to think of the Sun as orbiting the Earth) Notice also that the plane of the ecliptic is not perpendicular (i.e. at right angles) to the Earth's axis of rotation.  This is because the Earth's equator is inclined by about 23½ degrees to the plane defined by the Earth's orbit around the Sun.  (It is this tilt that gives rise to the sea­sons.)

When you look at the stars, you can see only half of the “visible sky.”  The other half is blocked by the Earth.  The limit of what you can see is called the horizon.  Therefore, the horizon ring represents the limit of what can be seen. 

 

Turn the knob that controls the Earth or rotate the celestial sphere instead.  (Remember that rotating the Earth-knob is equivalent to holding this knob and rotating the celestial sphere.)  You will notice that sometimes the Sun is above the horizon and sometimes it is below.  The time when the Sun is above the horizon is defined as the day, and the time when it is below the horizon is the night.  (Although we cannot see the stars during the day because of the brightness of the Sun, the stars are nevertheless still there in the sky.  For instance, at the time of a total solar eclipse, the stars will “come out” because the light of the Sun will have been blocked by the Moon.)  When the Sun is aligned with the eastern horizon, we have sunrise, and when the Sun is aligned with the western horizon, we have sunset.  Over the course of the day, the Sun appears to move from east to west across the sky.

 

Using the Earth-knob only, move the Sun east to west, from horizon to horizon.  Rotating the Earth-knob represents the daily rotation of the Earth on its axis.

 

When the sun-pointer knob is used to move the Sun, notice that the Sun passes in front of different stars on the celestial sphere.  These stars comprise the constellations of the zodiac, with which you are probably familiar.  The different positions of the Sun correspond to different times of the year.  The Sun will spend an average of about one month in each of the zodiacal constellations.  To understand this, think for a moment about the Earth’s revolution around the Sun and consider the diagram below.

 

Figure 2.1: The apparent change in position of the Sun relative to the stars

 

 

 

 

The Earth orbits the Sun once a year.  If we could see the stars during the daytime, we would see that on January 1st, for example, the Sun would appear to be positioned near Star A in the sky.  As the Earth moves around its orbit, the Sun would appear to move slowly along the ecliptic on the celestial sphere, and would appear to be positioned near star B on March 1st.  This apparent change in position is a consequence of the Sun’s apparent yearly motion, not its apparent daily motion of rising and setting.  In this way, the Sun would appear to move around the celestial sphere once a year, and each day it would be positioned near a different star. 

 

The path of the ecliptic is demarcated on the globe with small lines that indicate the apparent position of the Sun for each date during the year.  Turning the sun-pointer knob changes the date, which corresponds to the location of the Earth in its orbit around the Sun.  Adjusting the Earth-knob changes the position of the Sun relative to the horizon, but not relative to the stars.  Since the time of day corresponds to the posi­tion of the Sun relative to the horizon, turning the Earth-knob changes the time of day.  Noon is defined as the time (of day) when the Sun is on the meridian at a given location, which is also the time when it is at its high­est point above the horizon for that particular day.  The time between successive noons is divided into 24 hours.  An hour corresponds approximately to the distance between the lines that connect the north and south celestial poles.  These lines are analogous to meridians of longitude on the Earth. 

 

To locate objects on the celestial sphere, we use a coordinate system that is similar to the system of longitude and latitude used to describe positions on the surface of the Earth.  Right ascension (the celestial equivalent of longitude, abbreviated R.A.) is measured along the celestial equator.  Right ascension is generally measured in units of hours and minutes.  On the globe, lines of right ascension are indicated at intervals of 1 hour.  Declination (the celestial equivalent of latitude, abbreviated Dec.) is measured in degrees above or below the celestial equator.  A declination measure below the equator is preceded by a minus sign.  A declination measure above the equator is often preceded by a plus sign.  On the globe, parallels (i.e. lines) of declination are indicated at intervals of 15º. 

 

Table 2.1: Example Stars

 

 

R.A.

Dec.

Betelgeuse

5h52m

+7o

Vega

18h37m

+39o

 

 

3.) With the definitions given above, fill in the appropriate R.A. and Dec. for the stars listed in Table 2.2.  Remember to include the constellation to which the star belongs.

 

 

Setting the Globe for a Specific Geographic Location and Time

For the following questions, begin by setting the globe for “noon” at our location, which we will  consider as Washington, D.C.  Its latitude and longitude are as follows:

 

Latitude: 39º N             Longitude: 77º W

 

•        Rotate the celestial globe and the Earth globe until our location is "on top" (i.e. Washington, D.C. should face the ceiling.)  Degrees are marked on the meridian ring.  To orient the globe correctly, the mark indicating 39º should be positioned at the zenith. 

 

Note: No matter where you are on the surface of the Earth, you can consider yourself as being "on top;" the point directly overhead will be the zenith.

 

•        The directions North, South, East and West are indicated on the base (i.e. the stand) of the celestial sphere.

 

•        Position the Sun at today’s date.

 

•        Set the globe for 12:00 noon by holding the Earth-knob fixed and rotating the sky until the Sun is on your meridian.

 

Before proceeding, have your instructor check that your globe is set correctly.

 

4.) Which constellation is closest to the zenith at noon?

 

5.) Which named star is closest to the zenith at noon?

 

6.) What is the Sun's altitude at noon?

 

7.) What direction would you face in order to see the Sun at noon?

 

To set the globe for other times of the day, remember that the Earth turns 15o/hour from west to east (15o/hour x 24 hour = 360o, which is equivalent to a full rotation).  Equivalently, the whole sky moves 15o/hour from east to west.  It is easier to simulate the Earth’s rotation by holding the Earth-knob fixed and rotating the celestial sphere.

Rotate the celestial sphere westward until it is sunset (i.e. until the Sun coincides with the horizon ring).  As you rotate the sphere, count the number of lines of R.A. that pass under the meridian bar.  The number of lines of R.A. through which you have rotated the sky equals the number of hours it is after noon. 

 

8.) How many hours after noon does sunset occur?  (Give your answer to the nearest ½ hour)?

 

9.) What is the Sun's altitude now (i.e. at sunset)?

 

10.) What direction would you face to watch sunset today?  

 

11.) At what longitude is it now noon?

 

12.) Is the constellation you found in question 4 still above the horizon at your location?  (Indicate totally, partially, or not at all.)

13.) Mirach (in the constellation of Andromeda) and Markab (in the constellation of Pegasus) are two stars that are just rising at sunset.  Ask your instructor for the location of these stars on the celestial sphere and then complete Table 2.3.   (See the Appendix at the end of this write-up for instructions on how to measure altitude and azimuth.)

 

Now rotate the sky to 3 hours past sunset.

 

14.) What are the altitude and the azimuth of the Sun?

 

15.) Is the constellation from question 4 still above the horizon? (Totally, partially, or not at all.)

 

16.) Now what are the altitudes and azimuths of the stars in question 13?

 

Challenge Questions:

17.)    It is noon on the 20th of May.  You are on a sailboat and you use a sextant to measure the alti­tude of the Sun.  The Sun is to the south and you measure its altitude to be 75°.  What is your latitude?  Explain your answer.

 

 

18.)      You are on the same sailboat as in Question 17, but it is now July 5th and it is late at night.  You have just been through a storm and all of your maps have been washed overboard.  However, being brilliant and resourceful and having taken astronomy at GMU, you realize that all is not lost!  You look for the bright star Vega in the constellation of Lyra and find that it is at your zenith.  A chronometer (i.e. an accurate clock) on board indicates that it is noon in Greenwich, England.

 

            A.) In what body of water are you?

 

B.) What are your latitude and longitude?  Explain.  No credit will be given without an appropriate explanation.

 

 

 APPENDIX (Some additional information):

 

Zenith: The point on the celestial sphere that is directly overhead of your location.  Every point on the surface of the Earth has a different zenith, and the point on the celestial sphere corresponding to the zenith for any specific observer constantly changes with location and the time.

 

Meridian: An imaginary line on the celestial sphere drawn from the exact north point on your horizon up through your zenith and down to the exact south point on your horizon.  It divides the sky into an eastern half and a western half.  The meridian is constantly changing with respect to the celestial sphere as the Earth rotates.  It is on the meridian that the stars reach their highest point in the night sky.

 

Altitude: The angular distance measured vertically above or below the horizon to a given object.  The zenith has an altitude of +90°.  The horizon has an altitude of 0°.  Altitudes below the horizon are negative.

 

To measure the altitude of an object: Draw an imaginary curve, from the object being observed, directly down to the horizon (horizon ring in the case of the Farquhar globe).  Measure the number of degrees above the horizon.  Start with an altitude of 0º at the horizon and estimate degrees as you move upwards.  This same procedure can be used for an object below the horizon if you adapt it in the appropriate way.

 

Azimuth: The angular distance of a given object measured eastward around the horizon from the North (i.e. the exact north point).  The meridian has azimuth 0° for its northern half and 180° for its southern half. 

 

To measure the azimuth of an object: Draw an imaginary line from the object down to the horizon.  If the horizon ring on your celestial sphere is demarcated in degrees, then you can determine the value of the azimuth simply by reading the number at the point where the line you drew intersects the ring.  If your horizon ring is not marked, you can approximate the azimuth using the following information:

 

N = 0º

NE = 45º

E = 90º

SE = 135º

S = 180º

SW = 225º

W = 270º

NW = 315º