Abstract
One of the objectives of science is to understand the phenomena that we observe, and discover the underlying rules that create and govern them. To cite two examples: “I see the sun rise and set … why does it do that?” and “At night, I see the stars move across the sky … do they mimic the motion of the Sun?” By investigating questions like these, we hope to discover the underlying rules that govern how the physical world operates.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Microsoft MovieMaker is included with modern Windows operating systems. It is quite capable of handling this project.
- 2.
This statement assumes that you are located in the northern hemisphere, e.g. North or Central America, Europe, Asia). If you are located in the southern hemisphere (e.g. southern Africa, Australia, South Pacific) the paths will “tip” toward the north. If you are very close to the equator (e.g. Central Africa, Central South America) the paths will rise nearly vertically from the horizon.
- 3.
These definitions of “day”, whether the Sidereal Day based on the stars or the Solar Day based on the Sun, are distinct from the concepts of “daytime” and “night-time”. You know from experience that in a standard civil day of 24 hours, there is a period of “daytime” when the Sun is up, and a period of “night-time” when the Sun is down. Depending on the season and your geographic location, the interval of daytime may be longer or shorter than the period of night-time. In the formal definitions that we’ll use in Project 2 and Project 3, a day contains one interval of daytime plus the contiguous interval of night-time.
- 4.
The “range” of measurements can be described in many different ways. The most common is to calculate the average value (μ) and the standard deviation (σ) of the measurements. The standard deviation is a way of expressing the uncertainty in the value of the parameter that you are trying to determine (in this case, the length of the Sidereal Day). Under ideal conditions, the underlying “true” value of the unknown parameter being measured has a 68% probability of lying between μ – σ and μ + σ. An alternative, more conservative, way of expressing the uncertainty is to state the full range (±) of the measurements, such as “23.93 ±0.01 hours”. The standard deviation is the most common way of expressing measurement uncertainty, but both approaches are used in the scientific literature, so you will need to state clearly the meaning of your own measurement uncertainties. People often call the standard deviation the “measurement error”, but this is a misnomer. The normal statistical variation in a series of repeated measurements is not an error. It is not a mistake. Rather, it is an unavoidable property of technical measurements. Reading the clock incorrectly would be a genuine error. If you have made an error, then you recognize it, avoid it in the future, and make a correct measurement.
- 5.
The measurement uncertainty expressed here is the full range of the calculated length of the Sidereal Day, from a total of 9 event timings.
- 6.
There are commercial solar filters available at optical and astronomical suppliers. With such a filter you can safely observe the Sun, with and without optical magnification. They are specially designed and made for solar observation, and they are excellent tools for some purposes. Even with these devices, it is important that the observer understand the proper method of using them, and their limitations. You won’t need any of them for the projects in this book.
I caution you against the temptation to look directly at the Sun through welder’s goggles, smoked glass, film negatives, or the like. These are occasionally mentioned in old stories as “rough and ready” solar filters. The problems with them are:
(a) You don’t know the details of their design and manufacture, hence you don’t know if they will actually protect you from the visible, infrared, and ultraviolet rays of the Sun, any of which can cause serious visual damage.
(b) Just because you don’t see any glare, and don’t feel any pain, does not mean that you are safe. The ultraviolet and infrared radiations don’t trigger a visual sensation, they just burn your eye; and there are no pain sensors on your cornea or retina, so you may not feel the damage being done. If you ever meet a mountain climber, ask him about “snow blindness”, which is caused by overexposure to ultraviolet radiation (due, in that case, to a combination of high altitude that reduces the UV-protective effects of our atmosphere, and enhanced reflection of the UV off of the snow and ice surfaces). He will tell you that in many cases there is no pain, nor any other obvious warning sensations, until you simply lose the ability to see. Not good! Climbers wear dark goggles that are designed to protect them against the light reflected off of mountain surfaces. These goggles are not meant for looking directly at the Sun.
(c) Some of the “rough and ready” solar filter notions of yesteryear may have actually worked, but manufacturing technology has changed and with it the performance of such filters. Specifically, there was once a time when the type of silver-halide chemical left on fully exposed and developed black & white negatives could provide tolerably good protection for direct solar observing. However, as film chemistry changed, so did the UV and IR characteristics of developed film. I don’t remember the details of which film chemistry, suppliers, and manufacturing methods provided acceptable solar filtering, and neither do you. Don’t take the chance. Besides, when is the last time that you saw any black & white film negatives? Color film negatives, with their organic dyes, were never acceptable protection against UV or IR radiation.
I hope that you get the message here, and that you take it very seriously. Use only pinhole projection for the solar observations in this book. Don’t be stupid about solar observing, and do take care of your eyes.
- 7.
Refer back to the warnings about looking directly at the Sun, in Project 3. If you accidentally damage your camera by taking a picture of the Sun’s disk, that is too bad, but not irredeemable. If you damage your eyes, then that is catastrophe. Don’t do that!
- 8.
It is an enlightening experience to simply watch how things evolve, beginning about an hour before sunrise. You may see the first yellow glow of sunlight high in the treetops (or skyscrapers) long before the Sun clears the horizon. If the night was calm, you may feel subtle movement in the air as the Sun’s warmth encourages the morning breeze. The night-sounds of crickets and frogs will fade away, to be replaced by bird-calls and buzzing insects. The stillness may be punctuated by the early bird searching out his worm. As sunrise approaches, the golden glow that began so high on the treetops or skyscrapers will migrate slowly downward until, just about the time that it is level with you, the Sun blazes into view over your horizon. At least once, you should invest the time to watch, hear, and feel this pageant unfold!
- 9.
Most camera lenses create “pincushion” or “barrel” distortion in the image. This is unavoidable with lenses that provide the wide field desired for this project. One implication is that the scale factors (KW and KH) are not truly constant across the image. As a result, there will be some distortion in your calculations of the Sun’s azimuth. My experience is that the zoom lens that is included with most DSLR kits is remarkably good in this regard, even when it is zoomed out to its widest field of view. The lens distortion, combined with inconsistent aiming of your camera, will impose a more-or-less random deviation in your measured sunrise azimuths, amounting to plus or minus a few degrees. You can reduce the lens distortion by zooming in to a narrower field of view (longer focal length), but this is likely to force you to deal with the complication of using two or more reference points in order to map the full year’s migration of the Sun’s azimuth.
- 10.
The annual floods that were central to Egyptian agriculture for thousands of years are now controlled and metered out by the great Aswan dam on the upper Nile.
- 11.
The declination of the Sun ranges from +23.5 degrees to −23.5 degrees over the course of a year. You’ll measure this in Project 9. If your latitude is in the northern temperate zone, then the shadow of your gnomon will always extend northward. At the summer solstice, the shadow’s reach will be L− and at the winter solstice the shadow’s reach will be L+. If you are located in the southern temperate zone, then the gnomon’s shadow will extend southward from the base of your gnomon but its total span will be given by the same equation. If you are in the equatorial zone, then the shadow of the gnomon will extend both northward (L−) and southward (L+) from the base of the gnomon, at different times of the year. In that situation, Eq. 1.8 gives you its maximum northward and maximum southward extent. In all cases, if the time you select to make your shadow measurements does not put the average Sun position on the meridian, then the Analemma will be tipped east/west, and its total span will be somewhat longer than indicated by Eq. 1.8, so after you’ve calculated L, give yourself some extra space.
- 12.
If you made two sets of timings on each day, you should look at the time-difference between them. If your measurements were perfectly accurate, then the difference between the two shadow-crossing timings would be the same, every day. When I did this project, the time difference between shadow-crossing “A” and shadow-crossing “B” ranged from 2 minutes to 4 minutes, with a standard deviation of 35 seconds. This is probably about the best that you can expect to do with visual timings as described for this project.
- 13.
The terminology here can be a bit confusing until you get used to it. Astronomers and other experts on timekeeping (“horologists”) use the term “apparent Sun” to indicate the observable position of the actual Sun. The term “mean Sun” refers to a fictitious, mathematical construct that moves at a constant angular rate along the celestial equator. In this term, “mean” is a synonym for “average”. The angular rate of the “mean Sun” is the average angular rate of the real Sun, averaged over one year. The length of the year is defined as the time it takes the Sun to completely circumnavigate the sky, relative to the stars. It is roughly equal to 365.25 days (so the angular rate of the “mean Sun” is about 360 degrees divided by 365.25 days). We arbitrarily divide the day into 24 equal intervals.
- 14.
The meridian divides the celestial sphere into two equal halves, one eastward and the other westward. It is the line that starts in the North, passes directly overhead (the zenith), and continues to the southern horizon.
- 15.
With the “Go-To” mounts, you must always aim from target to target by using the hand control to drive the mount’s motors. If you “manually” change the pointing, you will disturb the initialization and synchronization of the mount, and the reported altitude-azimuth or RA-Dec angles will be erroneous.
- 16.
Historians and archeologists have not found any compelling evidence for the design of the instrument that Hipparchus used to make his measurements of celestial positions, but it must have been capable of making the same sort of measurements that you made using your equatorial theodolite. There have been some academic arguments about whether Hipparchus actually measured stellar positions in the polar-equatorial coordinate system; but most investigators seem to agree that his table of stellar positions is based on their coordinates in this polar-equatorial coordinate frame.
- 17.
Be careful with the subscripts and the time-arguments when you use this equation. It is worthwhile to derive this equation for yourself using Eq. 1.12, to be sure that you understand what Eq. 1.13 means. “A2 – A1” is the angular distance from star 1 to star 2 (in the alpha-direction). If star 2 is east of star 1, then A2 – A1 will be a positive number. The “measurements” are the actual readings on the alpha-scale of your theodolite when you aimed at star 1 and star 2.
- 18.
These “A, δ” coordinates differ from the astronomer’s standard RA and Dec coordinates only in the zero-point of the longitudinal coordinate. You’ll see the reason for the difference, and discover how to translate your “A, δ” positions into RA and Dec, in Project 9.
- 19.
The calculations will utilize Eq. 1.13 to compensate for the fact that the celestial sphere is continuously rotating about the polar axis. If you measure one star, then move to another star and measure it, you have two measurements at two different times and the sky will have rotated during the interval between the two measurements. The observations you’ll make for this project are very similar to those done by the ancient astronomers, but the calculations as described here are a bit different than what they did. For the modern replication, you’ll rely on an accurate clock. The ancients didn’t have accurate clocks, so some more clever (and laborious) methods had to be used to deal with the sky’s continuous rotation.
- 20.
The scale on some telescope mounts gives the alpha angle in “hours” instead of “degrees”. This tradition has a long history and the rationale for it is convoluted. Since the sky rotates one full circle (360 degrees) in approximately 24 hours (one day), astronomers divide the circle into 24 equal parts. The conversion is 1 hour of α-angle equals 15 degrees of α-angle. Smaller divisions are expressed in minutes and seconds (1 minute = 1/60th degree, etc.). The desired accuracy of ±0.5 degree of alpha angle is equivalent to ±1/30th hour of alpha angle, which is ±2 minutes of alpha angle.
- 21.
- 22.
Examples of naked eye aiming devices are a sight-tube attached to the telescope, or a Telrad® sight.
- 23.
The readout on most “Go-To” telescope mounts gives the Right Ascension angle in “hours-minutes-seconds” (HH:MM:SS.ss) instead of “degrees”. For most purposes – including plotting the stars on your celestial globe, it is convenient to convert the RA into decimal degrees, using DD = 15 ∙ (HH + MM/60 + SS/3600). The declination angle is usually given in “degrees-minutes-seconds”. You can convert this to decimal degrees by DD = DD + MM/60 + SS/3600.
- 24.
- 25.
The scale on some telescope mounts gives the alpha angle in “hours” instead of “degrees”. This tradition has a long history and the rationale for it is convoluted. Since the sky rotates one full circle (360 degrees) in approximately 24 hours (one day), astronomers divide the circle into 24 equal parts. The conversion is 1 hour of α-angle equals 15 degrees of α-angle. Smaller divisions are expressed in minutes and seconds (1 minute = 1/60th degree, etc.) The desired accuracy of ±0.5 degree of alpha angle is equivalent to ±1/30th hour of alpha angle, which is ±2 minutes of alpha angle.
- 26.
In the data reduction for this project, you will adjust the Reference star measurement to the corresponding time of a “Moon” measurement. You can’t do the inverse (time-adjust the Moon’s measured alpha angle) because the Moon isn’t attached to the celestial sphere. It doesn’t follow the sidereal rate (ω), and in fact (as you’ll find from this project) its rate of motion across the celestial sphere isn’t constant.
- 27.
Examples of naked-eye aiming devices are a sight-tube attached to the telescope, or a Telrad® sight.
- 28.
The readout on most “Go-To” telescope mounts gives the Right Ascension angle in “hours-minutes-seconds” (HH:MM:SS.ss) instead of “degrees”. For most purposes – including plotting the stars on your celestial globe, it is convenient to convert the Right Ascension into decimal degrees, using DD = 15 ∙ (HH + MM/60 + SS/3600). The declination angle is usually given in “degrees-minutes-seconds”. You can convert this to decimal degrees by DD = DD + MM/60 + SS/3600.
- 29.
The illustration in Figure 1-23 treats the celestial equator as the fundamental plane. However, in the case of the orbits of solar system objects (including the Moon), the normal astronomical convention is to measure the inclination and the nodes relative to the ecliptic, rather than the celestial equator. You will find out about the ecliptic in the next project.
- 30.
During a total solar eclipse, some stars will be observable. In this special, rare event it is possible to actually measure the Sun’s position among the stars, but it turns out to be a fairly difficult observation to make.
- 31.
That is, 0.04 deg/hr ∙ 24 hr/day = 0.96 degree/day.
- 32.
Check an almanac or www.SkyandTelescope.com to confirm that you don’t accidentally choose one of the bright planets as a Reference star.
- 33.
The scale on some telescope mounts gives the alpha angle in “hours” instead of “degrees”. The conversion is: 1 hour of α-angle equals 15 degrees of α-angle. Smaller divisions of the alpha angle might be expressed in minutes and seconds (1 minute = 1/60th degree, etc.) The desired accuracy of ±0.5 degree of alpha angle is equivalent to ±1/30th hour of alpha angle, which is ±2 minutes of alpha angle.
- 34.
For example, if you measured the Reference star at tR = 2014 July 25 at 8:30 PM, and you measured the Sun on the next day, at tS = 2014 July 26 at 2:45 PM, the time difference is [tR – tS] = −18.25 hours.
- 35.
The readout on most “Go-To” telescope mounts gives the Right Ascension angle in “hours-minutes-seconds” (HH:MM:SS.ss) instead of “degrees”. For most purposes – including plotting the stars on your celestial globe, it is convenient to convert the Right Ascension into decimal degrees, using RA = 15 ∙ (HH + MM/60 + SS/3600). The declination angle is usually given in “degrees-minutes-seconds”. You can convert this to decimal degrees by Dec = DD + MM/60 + SS/3600.
- 36.
Why “perhaps seven”? For quite a while it was not clear whether the morning and evening apparitions of Mercury represented two different objects, or a single object. Similarly, it wasn’t clear whether the morning and evening apparitions of Venus represented one, or two, objects.
- 37.
Viewed in this way, the ecliptic represents the plane of the Earth’s orbit around the Sun.
- 38.
Ptolemy probably used ecliptic coordinates because the motion of the planets was such an important topic of research in his era. Once the ancient stargazers began plotting the Sun’s path – the ecliptic – on their celestial globes, they realized that all of the astrologically significant objects that moved across the celestial sphere (the planets, the Moon and the Sun) stayed on or near the ecliptic. For calculations involving these bodies, it was convenient to describe their positions relative to the ecliptic plane, instead of employing the celestial equator as the fundamental plane.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Buchheim, R.K. (2015). Motions and positions in the sky. In: Astronomical Discoveries You Can Make, Too!. Springer Praxis Books(). Springer, Cham. https://doi.org/10.1007/978-3-319-15660-6_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-15660-6_1
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15659-0
Online ISBN: 978-3-319-15660-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)