Location and Time for Objects and Events Off of the Earth


As Douglas Adams once observed, "Space is really big." As such, if one was to design a coordinate system to describe the location of objects, one might end up with one of a variety of systems depending on how much of space one was interested in and what aspects of location one was interested in.


Planetarium Software There are several different software packages available. The best free option is, I think, Stellarium. I strongly encourage you to download it if you have not already done so. It will be useful on a few occasions in our course work, but mostly it is a great way to enhance your exploration of the night sky.


Finding Objects on the Sky: Celestial Co-ordinate Systems
This involves knowing where to look relative to the stars. Most stars maintain fixed positions relative to each other. Even if they are moving through space relative to each other, they are so far away that this motion is not significantly visible to us. The Earth, on the other hand, rotates on its axis and revolves in its orbit. So what stars we see if we look in a certain compass direction and at a certain angle up from the horizon depends on the time of day and the position of the Earth in its orbit. So, it makes sense to anchor our sky co-ordinate systems to the stars rather than the Earth.

Since, from our perspective, the sky seems to form a sphere around the Earth, a co-ordinate system similar to latitude and longitude is most useful. These co-ordinate systems do not include a third term for distance (analogous to elevation) since we are only using them to aim our line of sight on the sky, not to plan a voyage. The co-ordinate systems usually are centred on the Earth because that is where we are. Some are centred on the sun. Basically, these systems imagine that all objects in the sky are drawn on a celestial sphere surrounding the centre of view.

To be clear, these co-ordinate systems give the position of the objects on the imaginary celestial sphere! As the earth rotates under this sphere, objects will change their position on your sky. A particular latitude and longitude location on the surface of the earth does not correspond to a particular celestial position.



Equatorial Co-ordinates
This celestial co-ordinate system closely matches the latitude/longitude co-ordinate system used to locate objects on the earth's surface. It uses two co-ordinates.

Declination (dec): celestial latitude.
The "equator" for this system is a projection of earth's equator up onto the celestial sphere. If you are on the equator then the celestial equator passes directly overhead (a point on the sky called the Zenith). The poles on equatorial celestial sphere are easy to spot. One need only take a time-lapse photograph. The stars in such a photograph all trace out circles centred on the celestial poles.

Figure 1: star trails in a roughly 6 hour exposure.
Over that time, the camera captures enough light from the landscape to look like daylight.


This is an image of the south celestial pole which is at dec -90o 0' 0". However, the photograph was not taken anywhere near Earth's south pole. How do we know this?

Measuring Declination
Declinations south of the equator are negative, those to the north are positive.

Fractions of a degree could be given as a decimal, but this is not how it is normally done.
Instead we use:
arc minutes (60 arc minutes per degree), and
arc seconds (60 arc seconds per arc minute).

For example, a star with a declination of 45o 12' 14.25" would be read as
45 degrees,
12 arc minutes, and
14.25 arc seconds.

To convert to decimal degrees: 45 + (12/60) + (14.25/3600) = 45.2039583o.

Since this declination is similar to our latitude in Ottawa, this star will be be directly overhead (at our Zenith) once a day as the Earth rotates.


Right Ascension (RA): celestial longitude.
The name comes from the fact that when standing on the equator and looking east one sees objects ascend up from the horizon along paths that are at right angles to the horizon. If one were to draw lines of equal right ascension on the sky they would run from pole to pole. Starting from the zero RA line, RA increases to the east as seen from Earth. So if you have a particular star directly overhead and a friend of yours who is off to the east is looking at a different star that is directly overhead of them, then their star has a greater RA co-ordinate than yours. However, since Earth rotates from west to east, if you wait long enough the earth's rotation will bring your friend's star to a point on the sky directly above you, as long as they were at your latitude.

Figure 2: RA and dec lines as they would appear looking south
from Ottawa in early winter. Constellation lines also added. From Stellarium.



Measuring RA
RA, unlike longitude, is not measured in degrees. It is measured in hours.
There are 24 hours in a full 360o circle, so 1 hour of arc is the same as 15o. RA increases eastward across the sky. So if you are looking up and to the south at a star that has an RA of 5 hours (5h) then a star with an RA of 6h would be off to your left and a star with an RA of 4h would be off to your right. A star with RA 11h would be far to the left near the eastern horizon and a star with RA 23h would be far to the right near the western horizon. Approximately 1 hour later, the RA 6h star will be up and to the south. Approximately 5 hours after that the RA 11h star will be high in the sky to the south. I say "approximately" because an hour of Right Ascension is not the same as an hour of clock time. An hour of RA is 1/24 of a sidereal rotation, but that takes only 23 hours and a bit over 56 minutes.
Just remember that a Right Ascension hour is a measure of angle, not time.

Fractions of an hour of RA are measured minutes and seconds.
60' per 1 hour
60" per 1'
An RA of 4h 32' 15" would be read as 4 hours, 32 minutes, 15 seconds.
This would be the same as 4 + (32/60) + (15/3600) = 4.5375h of RA.

RA vs dec Units
1' of Right Ascension is not the same angle as 1' (arcminute) of declination. This is because an hour of RA is 15o, so 1' of RA is a 15 times bigger arc than 1' of declination. The same is true of seconds of RA vs arcseconds of declination. Irritating, but if you just work with the given units and don't try converting, you don't need to worry about these differences in practice.

Ra Hours vs Clock Hours
I will repeat: one hour of Right Ascension is not the same as an hour of clock time. Remember that a sidereal day (a 360 degree rotation) takes 23 hours, 56 minutes, and 6 seconds. So it takes less than 1 hour of clock time for the sky to shift sideways by 1 hour of Right Ascension: 59 min and 50.25 s.

The Gap between RA Lines
The other important consideration for RA is that the lines come together at the celestial poles and are farthest apart at the celestial equator. Thus for stars with a declination near zero, a difference in position on the sky of two stars 1h apart is greater than for two stars 1h apart near the celestial poles.

Figure 3: RA and dec lines as they would appear looking north
from Ottawa in early winter. Constellation lines also added. From Stellarium.




Zero RA Line
As with longitude, the trick with measuring right ascension is to pick a precisely definable zero point for your measurement. The zero point that astronomers have picked is the point on the sky where the sun's path across the celestial sphere (the ecliptic) crosses the equator (marking the spring equinox). This spot in the sky is called the "First Point or Aries," but it actually is in the constellation Pisces. The name comes from the fact that when the Greek astronomer Hipparchus first identified this point in the sky in about the year 130 BCE it was just inside the constellation Aries (the point where the sun first enters that constellation). As we learned in our investigation of time, the slow wobble of the Earth's rotation axis means that the point of the Vernal Equinox shifts westward by about 1 degree every 72 years. If you are curious how we know where in the sky the sun is relative to the stars when its path crosses the celestial equator, an amateur astronomer named Patrick Powers has a nice explanation.


Figure 4: Summary of RA and dec angles as they are defined on the celestial sphere.
From Wikipedia.
RA is shown in 1 hour increments, dec in 15o increments.
The star depicted in this image is at RA 5h and dec 52o 30'

The cage formed by the celestial sphere co-ordinate lines stays in a fixed orientation with respect to the stars while the Earth rotates inside it.

Note: the ecliptic is the path of the sun on the celestial sphere.



Figure 5: Looking up from Earth's Surface at the celestial sphere.
The wide angle of view increases the distortion of the co-ordinate lines.
Equator (zero dec) is yellow and zero RA line is orange. From Stellarium.


The stars and Milky Way (whitish band) will appear at the same co-ordinates each night. The moon, sun, planets, and other minor bodies within the solar system change their co-ordinates (mostly along the ecliptic) from one observation to the next.

Even the stars and galaxies change their celestial co-ordinates as they and we move through space. Their distances are such that the effect of this is only apparent with very sensitive instruments and/or very long time spans between observations.

As time passes over the course of a night, the Earth's west-to-east rotation will cause each star in this image will move across the sky along a path parallel to its declination line resulting in a set of concentric, counter-clockwise arcs around the north celestial pole similar to the photograph in Figure 1. The RA co-ordinate lines will move along with them.
Figure 6: How RA and dec are used to aim telescopes.
Equatorial celestial co-ordinates are mostly used for aiming telescopes rather than for unaided-eye astronomy.

Telescope mounts have two perpendicular axes of rotation to aim them.

The declination axis:
* aims the telescope at a particular declination on the celestial sphere

The polar axis:
* is lined up with the north celestial pole
* rotating on this axis aims the telescope at a particular RA line on the celestial sphere

Once the telescope is properly aimed, it has a motor that can rotate the rest of the mount around the polar axis at the same rate as the earth rotates, but in the opposite direction. This keeps the telescope aimed at the same spot on the celestial sphere as the earth rotates.

The elegance of this system is that the telescope is easy to aim and keep aimed by just one motor.



Figure 7: Beta Cassiopeia and finding Zero RA in the sky.
While it is easy to find the north celestial pole if you can find the little dipper, there is no easy-to-see star marking 0h RA.
An easy to spot start that is reasonably close to 0h is beta Cassiopeia (the second brightest star in the constellation Cassiopeia). It is at RA 0h 09' 0.6" and dec 59o 09' 13.1". Cassiopeia looks like an "M" or a "W" depending on where it is as it circles around the north star (Polaris). One side of the "M" is a wide-open angle, the other is a narrow angle. Our target star is the end star at the narrow-angle end.

Trace an arc from the north star through beta Cassiopeia and through the side of the box-shaped constellation called Pegasus (the side closest to Orion). This arc is the RA 0h line. Find Polaris, Cassiopeia, and Pegasus in Figure 5 above.

Note the vertical streak to the left of the image. This probably is caused by a meteor.

Measuring the RA of a star on the celestial equator when you are on Earth's equator is fairly easy. It is given by what time the star rises above the horizon relative to a star at zero RA. Imagine that you see such a star rising up from the horizon and you recognise it as being at 4h RA. Any star that rises up 59 minutes and 50 seconds later must have an RA of 5h. If you are not on the equator, you need to be really comfortable with estimating angles and with spherical trigonometry to be able to make such a determination.

No matter where you are, wait until a star whose RA you know is overhead, due south, or due north of you. Draw a straight line through that star and the north or south celestial pole. All stars on that line have the same RA.


For the most part, unaided-eye astronomers ignore celestial co-ordinates and instead they communicate where in the sky an object is by stating in what constellation it is. Unaided-eye astronomers quickly get good at recognizing constellations.

To get practiced at this:



Precession
As noted above, the Earth's rotation axis wobbles. It completes one cycle of this wobble every 26,000 years or so. Thus, the direction on the celestial sphere that the rotation axis points to keeps changing, inscribing a circle before coming back to point at the Polaris again 26,000 years from now. Back when the great pyramids were being built in Egypt, the celestial north pole was about half-way between the end two stars in the tail of the constellation Draco the dragon. 12,000 years from now, it will point to the star Vega in the constellation Lyra.

Precession also changes the orientation of the equator. Since the orbit of Earth around the sun is not changing, precession causes the equinoxes to happen a bit earlier each year.

It is important to note that the location of the equator and the poles on the earth is not changing (much). It is the orientation of the earth in space that is changing.


Precession means that our frame of reference for celestial equatorial co-ordinates is constantly changing. So why did we use the vernal equinox as a reference to define our co-ordinates if it always is changing? The reason is that everything else in space is moving around as well. And the farther away an object is, the less certain we are about how it is moving. Using the sun and the earth uses two close-by objects, so we get good precision, at least.

How do we deal with the shifting equinox?
Every 50 years we reset the co-ordinate system to the new position of the spring equinox on the celestial sphere. We note that position relative to distant stars, which do not move significantly over that time period. The 50-year period is referred to as an Epoch. To make sense of a set of co-ordinates, we need to know in what epoch they were recorded. There is lots of fun math to convert between epochs.

Our current epoch is J2000 which means the vernal equinox that we use to fix co-ordinates is based on the orientation of the Earth's axis as of 12 noon, January 1st of the year 2000 in the Julian calendar (about 13 days off from our Gregorian calendar).

Do we need to worry about this? No, but it is interesting to see what tricks the astronomers need to employ just to specify where something is in the sky with any precision.



Ecliptic Celestial Co-ordinates
Figure 8: Ecliptic co-ordinates of a star close to that in Figure 4.
From Wikipedia
Rather than use Earth's equator and north pole as our frame of reference, we could use the plane of the ecliptic (the plane in which the Earth orbits the sun) instead. There is no particular advantage to this system over equatorial co-ordinates. The vernal equinox is still the reference for zero declination and right ascension, so precession is still a problem.

Since equatorial and ecliptic celestial co-ordinates are at a small angle to each other, there is some fun math to convert between them.

This system is still centred on the Earth. If we ever start moving around in the solar system in a serious way in the future, then an ecliptic co-ordinate system centred on the sun combined with a distance from the sun co-ordinate could be useful for navigation.

For us earth-bound observers, ecliptic celestial co-ordinates are not of much use.




Galactic Celestial Co-ordinates
Figure 9: Galactic Celestial Co-ordinate System (still Earth-centred)
 Figure 10: Plane of the Milky Way


Because our galaxy is very thin compared to its lateral extent, an earth-centred galactic co-ordinate system only needs to specify a direction relative to the centre of the galaxy and a distance. It is a useful system to specify where parts of the galactic disk are relative to each other. For example, it shows us the distance from our sun to the galactic centre. It also shows us that we are on the edge of a minor arm in the messy spiral structure of our galaxy.

Neither of the other two systems does a good job representing the galaxy's structure because both earth's equator and the plane of the ecliptic are at very steep angles to the disk of the galaxy.

How we know about this galactic structure and our place in it is a question for later in the course.



Figure 11: Relative orientations of the Earth, Solar System, and Galaxy

Constellations 3 and 4 in Figure 11 represent the direction of average rotation (3) and its opposite for the region around our solar system. The actual direction toward which our solar system is moving is the constellation Hercules. The bow-wave made by our solar system in the inter-solar gas and plasma (low in density as it is) is almost directly towards the galactic centre. There must be a relatively high-speed wind in this material emanating from the galactic centre.



Figure 12: Relative orientations of the Earth-centred north poles for the earth (blue circle, also known as the rotation axis),
the ecliptic plane (red circle), and the galactic plane (green circle). The red line is part of the ecliptic plane.
The green 'X' is the mean galactic rotation vector. The wide, green band is the disk of the Milky Way.
We see the Milky Way disk as a band across the sky because we are inside it.
This is the view looking north at Ottawa's latitude at around 10 PM in September.




Figure 13: Detailed Map of the Celestial Sphere





Distance

Measuring distances within the solar system is tricky enough. This is true for both practical as well as conceptual reasons (some of the latter of which may become apparent in the next section). None-the-less, a little geometry combined with some radar measurements go a long way to figuring out how far away things are at this scale. For more distant objects, we have a bit more to learn before we can understand distance adequately. We will deal with the topic of measuring distance, in its entirety, later in the course.





Time

If you watched the PBS Spacetime video on the subject of spacetime you already will suspect that assigning a time co-ordinate to distant events is not going to be a trivial task. There are several reasons for this.

1) The Speed of Causality
The fastest speed at which any event can reach out and effect anything else is "c," the speed that we usually call the speed of light: exactly 299792458 m/s. It is an exact value because the metre is defined from the speed of light. No causal effect of any sort travels an faster than this. As a result, any event that you view at a distance actually happened at some time in the past. If you observe a massive flare on a star 1 light year distant, the event actually took place 1 year ago. The same is true for radio messages from distant civilizations. This limits the degree to which civilizations in different solar systems can stay meaningfully connected. If the distance is great enough, the response to any message you send will be received by your descendants rather than by you. Whether that response has any significance to them is not guaranteed.


Some consequences of this are beneficial. The finite speed of causality allows us to look back in time when we look off into the distance. This allows us to understand the history of the universe much better.

The finite speed of causality does mess with our concept of simultaneity, however. We don't see or feel the effect of distant events as they happen "now." For example, the star Betelgeuse is about 500 light years away, so we see it as it was 500 years ago. Whether it even exists now we cannot know (it is due to explode within the next few tens of thousands of years) and if it does, we won't know for 500 years. In some sense, Betelgeuse's 500-years-ago past can be thought of as part of our present, since only those events on Betelgeuse more than or equal to 500 years ago can have any affect on us here on Earth.

2) The "Passage" of Time
As the PBS video pointed out, the Special and General theories of relativity predict that two systems that start out with synchronized clocks will NOT have those clocks stay synchronized if they are moving relative to each other or are in significantly different situations. The Twins "Paradox" is an example of this effect (quotation marks because it is not really a paradox).

Acceleration, including that due to gravity, causes one to experience fewer time intervals than a system experiencing a weaker gravity. This has nothing to do with how the clocks are constructed or how precise they are. It is the behaviour of time itself that is affected by gravity, not the clocks. As a practical example, the clocks on the GPS satellites run faster than the clocks down on earth where the gravity field is slightly stronger. The US Air Force must use a correction factor in their calculations for the GPS system or the receivers would accumulate errors in their readouts on the order of kilometres each day.

This has practical implications for sending robots and people on deep space missions. In an extreme case, astronauts could return home to find that their children are older than they are. It also means that there really is no sense of absolute time across the universe with which to synchronize events taking place in it, at least until we embrace the general relativity concept of spacetime.

3) Sequences of Events
As the PBS video pointed out and as the preceding two sections suggest, it is entirely possible that two observers might not agree on the order in which a set of events took place even if both record a perfectly accurate account of what they observed. The Ladder "Paradox" is a good example of one way that this might happen. These discrepancies in observed sequence should only occur when the events in question are not causally connected (their separation in light years is greater than the their separation in years).


In Conclusion
What does all this mean for us when we look up at the sky? Almost all of what you see reflects a time before you were born. Many of the things that you can see no longer exist, but the effect of their demise has yet to reach us. If you want to put a fun message in a bottle for yourself, go out at take a look at the star Castor, the second brightest star in Gemini (actually a solar system with three pairs of stars orbiting each other). The light that Castor is emitting at that moment will not reach your eyes for nearly 52 years, around the time that you likely will retire from whatever career you choose to embark on. On that night, 52 years from now, go back out and take another look at Castor and the light that has been racing towards you for all those decades and reflect on all that you have accomplished during that time.