Motion in the Heavens

 

Some ancient cultures explained the nightly motion of the stars across the sky as the result of the rotation of a vast Celestial Sphere in which the stars are "studded" like diamonds. We still retain the term celestial sphere to help us define the apparent location of objects in the sky but we now understand that the nightly motion of the stars is due to our motion on a huge, spinning sphere - the Earth. In this section we will study the most fundamental aspects of the motion of the stars. In later chapters you will learn about the subtle motions of the sun, moon and planets
Figure 2.4 Roll-over image showing the Earth's rotational axis, orbital plane and celestial sphere.

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The Celestial Sphere

The celestial sphere is still a useful idea - we use it in a way analogous to how we use the surface of the earth to assign positions to objects in the sky. Figure 2.4 helps to introduce this idea. One way to think about the celestial sphere is to view as a projection of the earth out into space. Just as the earth has an equator, the celestial sphere is separated into North and South by the celestial equator.

Figure 2.4 shows the Earth's rotational axis as an imaginary line running through the earth and pointing in the direction of the North Celestial Pole (NCP). As the earth rotates around this axis we see the stars traveling in circular arcs around this point. If you are in the southern hemisphere then the stars appear to travel in arcs around the south celestial pole.

How to Define Position on the Celestial Sphere

In the science of astronomy it is vital that we have a method to define and describe the position of stars and planets. There are two commonly used methods:

The Navigational Method uses the terms altitude and azimuth to express position as an angle in the sky. Altitude is a measure of how many degrees an object is above the horizon. An altitude of 0 degrees is on the horizon while 90 degrees is directly overhead. Azimuth is an angular measure along the horizon from due north to east, south, west and back to north. An azimuth of 90 degrees is a 90 degrees clock wise rotation from due north - or due east. An azimuth of 180 degrees is due south and 270 degrees is due west. Although easy to visualize the Navigational method has the serious drawback that the altitude and azimuth of a star is constantly changing as the star appears to move in the night sky.
The Astronomical Method is very similar to how we use latitude and longitude to locate objects on earth. The position (in degrees) of an object above or below the celestial equator is called declination and it is the analog of latitude. Polaris, for example has a declination of 89 degrees - it is almost on the NCP (which by definition is 90 degrees up from the celestial equator or at a declination of 90 degrees). East-West position is called right ascension and is measured in hours, minutes and seconds from a specific point in the constellation Aires. Right ascension is analogous to longitude on the Earth. The astronomical coordinates are "attached" to the celestial sphere and move with the stars during the night. Thus, unlike the navigational method, the astronomical position of a star does not change during the night which makes it a very good way to define and describe the position of an object.

Figure 2.5a,b shows both methods applied to the bright star Sirius. In this figure the yellow boxes draw your attention to the positional information. In Figure 2.5a Sirius was at an azimuth of 163 degrees (south-east part of the sky) and and altitude of 18 degrees above the horizon. Figure 2.5b also shows that Sirius has a right ascension of 6 hours, 45 minutes and a declination of -16 degrees 43 minutes. This means that it is below the celestial equator.

 
Figure 2.5a Two different ways to define the position of the bright star Sirius in the sky. Figure 2.5b Two different ways to define the position of the bright star Sirius in the sky.

Where is the Celestial Equator in Your Sky?

By definition, the celestial equator is a projection of Earth's equator onto the celestial sphere. Figure 2.6a shows this as an imaginary line across the sky while Figure 2.6b shows how the altitude of the celestial equator is related to your geographic latitude.

Figure 2.6a The celestial equator shown in a winter sky. Figure 2.6b Diagram showing how the altitude of the celestial equator is related to an observer's geographic latitude.

A simple "thought experiment" will provide you with a rule of thumb to determine where the Celestial Equator is in your sky. The following example illustrates this:

Example 2.4 Where would the celestial equator be for: a) an observer located at the equator b) an observer at the north pole? How is the altitude of the celestial equator related to the observer's geographic latitude?

Solution: a) if you are the equator then the projection of the equator would be straight up - at an altitude of 90 degrees from your southern or northern horizon. Your geographic latitude is 0 degrees at the equator. b) If you were located at the North pole (90 degrees latitude) the celestial equator would be on your horizon at an altitude of 0 degrees. How are your geographic latitude and altitude of the celestial equator related? - They are complimentary angles (or, two angles which add to give you 90 degrees).

Figure 2.6b provides a simple way to determine the altitude of the celestial equator - it is just the compliment of the observer's latitude. As you will see in Chapter 3, knowing the location of the celestial equator allows you to determine the position of the sun during the winter and summer months.

Using Stellarium to Visualize Motion Around the NCP

Figure 2.7 shows a Stellarium simulation of the northern sky as seen from Edmonton. Note a number of things:

  • Everything appears to rotate around a point very close to the bright star Polaris. This is why we call Polaris the "North Star" or the "Pole Star". It currently lies very close to the North Celestial Pole and is therefore a very reliable way of determining the direction of true north. The astronomical coordinate system of declination and right ascension is shown briefly in the animation as a series concentric circles and arcs radiating from the NCP.
  • Some stars do not set! In fact you have quite likely noticed this with the bright star Capella, visible low in the north on a warm July of August evening. Stars that do not set are called circumpolar. Whether or not a star is circumpolar depends on both your geographic latitude and the position of the star relative to the NCP.
Figure 2.7 Stellarium simulation of motion around the NCP

You are strongly encouraged to download a copy of Stellarium and use it to help visualize the following examples:

Example 2.5 Where would you be on the earth if you looked straight up to see the North Star? Describe the paths of the stars in the night sky as seen from this location.

Solution: If you looked straight up to see Polaris then you must be at the North Pole! It is quite show that your geographic latitude is the same as the altitude or angle of the North Celestial Pole above the north horizon. All the stars would rotate around this point and this would mean that they would travel parallel to the horizon.

Example 2.6 Is there a "South Star"?

Solution: No! Figure 2.8 shows what the southern horizon looks like from the tip of South America. There are no bright stars within 20 degrees of the South Celestial Pole and hence no "Pole Star" for the southern hemisphere at the present time.

Figure 2.8 The southern horizon as seen from the tip of South America

Example 2.7 Use Stellarium to try and discover the rule of thumb that will tell you if a star is circumpolar or not as seen from any give location in Canada.

Solution: Figure 2.9 shows the constellation Cygnus the swan skimming the northern horizon. The star Sadr (g Cygni) just touches the northern horizon. Since the declination of Sadr is +40 degrees it is 40 degrees away from the North Celestial Pole. The latitude of Medicine Hat is 50 degrees North. There is a special connection between these two numbers! These numbers add to 90 degrees - they are complementary angles. This provides the following rule of thumb:

 
Any star with a declination greater than the compliment of a location's latitude will be close enough to the NCP to never set - it will be circumpolar. Figure 2.9 Northern horizon seen from the same latitude as Medicine Hat, Alberta. The star Sadr just skims the northern horizon - it is circumpolar from this location.

 

Sky Terms You Should Know:

Celestial Equator: the projection of Earth's equator onto the Celestial Sphere. It plays a role very similar to Earth's own equator in helping assign position to objects on the Celestial Sphere.
Celestial Pole: the direction in which the Earth's rotational axis points. The bright star Polaris is very close to the NCP.
Altitude: the angle of an object above the horizon.
Azimuth: the angle measured along the horizon, in a clockwise rotation from due north.
Declination: the angle an object is above or below the Celestial Equator
Right Ascension: The angle an object is along the Celestial Equator and measured from a starting point in the constellation Aires.
Zenith: directly overhead; straight up
Circumpolar: stars close enough to the NCP to be visible year round (latitude dependent).

Precession

But I am as constant as the northern star,
Of whose true-fix'd and resting quality
There is no fellow in the firmament.
The skies are painted with unnumber'd sparks,
They are all fire and every one doth shine,
But there's but one in all doth hold his place:

(William Shakespeare, Julius Caesar, Ac III, Scene 1)

Any rotating object, when tugged by outside forces will wobble. Our Earth is no exception and the combined tugs of the sun. moon and other planets produces a gentle wobble that causes the direction of the earth's rotational axis to wobble or precess with a period of 25, 800 years. This means that the current "north star" which is Polaris is actually changing. In another 8 thousand years or so the star Deneb will have the honor of being the "north star", in 12 000 years time it will be a truly magnificent sight when Vega plays that role. The following two movie clips should help display this. Figure 2.10 shows a small toy top spinning on a desktop. The shaft of the top is its rotational axis and as you watch you will notice a slow circular motion as the the shaft rotates around the point of contact with the table. As the top slows this wobble or precession becomes much more pronounced.
Figure 2.10 Video clip showing a top precessing on a desktop
Figure 2.11 Animation showing the path of the NCP with respect to the background stars.

Figure 12.11 shows the affect that the wobble in the Earth's axis has on the location of the NCP.

Example 2.8 When astronomers record the coordinates of a star or object they also must very carefully note the exact date on which the coordinate assignment was made. Why is the date important?

Solution: There are many reasons to record information such as the time of an observation but one very good reason is that the wobble or precession in the Earth's axis shifts the coordinate system (right ascension and declination) relative to the background stars. With extremely faint objects a failure to account for this may mean that the next astronomer, perhaps a decade later who tries to locate your object may not be able to! All modern position assignments are accompanied by an epoch or date for when the assigned coordinates were correct. The astronomer can then precisely account for the affects of precession. The following figure illustrates this - the coordinate of a star is preceded by "(J2000)" which is a short hand way of saying the coordinates were accurate on January 1 2000.

 

Practice

(Note - you may need Stellarium to help answer some of the following questions)

  1. Imagine that after a shipwreck you were washed ashore on a desert island. One the first clear night you notice Polaris low in the north. How could you determine the latitude of your new "home"?
  2. Describe the motion of rising and setting stars as seen from the equator - how would it appear different from how stars rise and set in the prairies?
  3. Look up your latitude (Google Earth would be an easy way to do this). Is the star Vega circumpolar from your latitude?
  4. How would the location of Polaris change as you travel southward?
  5. Where would you be located if there were no circumpolar stars?
  6. What is the altitude of the celestial equator for an observer in Calgary?

The understand the apparent motion of the stars at night

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The altitude of Polaris is very nearly the same as your geographic latitude

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Not only is Polaris NOT "true-fix'd" nut neither is it constant! Polaris is an example of a Cephied Variable star and changes its brightness by a few percent over a period of 3.97 days