04-14-2022, 09:03 PM
I have many books on celestial navigation and none of them explain the simple elegance of great circles as used in c-nav. That is a shame because they are easy to understand without any fancy mathematics whatsoever. I personally didn't “get” celestial until I figured out for myself how these great circles work. That is why I am posting this with the hope that some of you will find the following helpful – or at least interesting.
Many of us started out in celestial with the analogy of a flag pole and how the top of the flag pole appears at lower angles the further we stand from its base. This is a flat-Earth analogy that can only take you so far and sooner or later runs into limitations.
To begin with we must understand that all of the stars are so far away that the direction in which we see any stars doesn't change with our location on Earth.
For the moment consider Polaris, which seems to stand nearly still as the other stars rotate around it. If the entire Earth were the size of the period at the end of this sentence at that scale Polaris would be many times further away from the ink dot Earth than the Eiffel Tower is from New York City's Central Park. Clearly taking the bearing from one side of the ink dot in Central Park to the Eiffel Tower and then moving all the way to the complete opposite side of the ink dot and taking it again would not change the bearing any measurable amount.
In fact the real distance to Polaris is so far that even if you wait six months for the Earth to reach the opposite side of its entire orbit around the sun, the direction to Polaris won't change. What causes the change of the angle at which you see Polaris is a change in what you perceive as “horizontal” or “straight up” as you move about on the spherical Earth.
If we could transport ourselves far out into space and look back at Earth to see three people taking simultaneous observations of Polaris we would see that they are all looking in the exact same direction – at a star that appears to us to be directly over the Earth's north pole but which is incredibly far away. From our outer space location we would see that all three would be gazing northward in a direction parallel to Earth's polar axis. But Paul at the north pole sees the star directly over his head; Eddie at the equator sees it low on his horizon; and Matt in the mid latitudes sees it part way up his sky. What is different for each of them is what they consider to be “straight up” and “horizontal” depending on where they are standing on the sphere that is Earth.
This is true for all the other stars as well. It is only the spinning of the Earth that gives them the appearance of motion. If, from our outer space location, we took a look at several people taking simultaneous observations of Arcturus we would see that they too would all be looking in the exact same direction at that star, but the heights at which they each perceive the star in their local sky would differ by their locations on the sphere that is Earth.
Arcturus travels over the Big Island of Hawaii each day, so let us say we watch a bunch of people taking sights of Arcturus just as it is directly above Hawaii. To get a good mental picture of this imagine that you have a globe in your hands and hold it so that Hawaii is straight up. Now from your perspective all of the observers will be looking “straight up” at your ceiling and far beyond to see Arcturus regardless of where they are standing on the globe. Of course the further they are on the globe from Hawaii the more their personal sense of horizontal will be tipped over. If a few of them observe Arcturus at the same angle up in their sky we could say that they were all standing on a circle of position determined by the same distance from the star's geographic position. Alternately we could say that they are all standing on the same circle of position determined by an equal amount of horizontal tilting from the 0° tilt that occurs at the geographic position of the star.
Now on to great circles.
A great circle is a specific concept in spherical geometry. It is the circle that is formed on the surface of the sphere by a flat plane that passes through the exact center of the sphere. All great circles on any particular sphere have their centers at the center of the sphere. They all have the same diameter as the sphere, and they all have the same circumference, which is the maximum circumference of the sphere.
Think of cutting an orange exactly in half with a sharp straight knife. The outer edge of the peel on each of the two halves is now a circle. If the cut went through the exact center of the orange this circle would be a great circle on the orange. If instead that sphere were a beach ball it is a great circle on the beach ball; If the sphere were a classroom globe that circle would be a great circle on that globe; If the sphere were the Earth it is a great circle on Earth.
Any plane that does not go through the exact center of the sphere still makes a circle where it intersects the surface, but that is not a great circle. Instead it is technically a “small circle” even if it is enormous in size. All lines of latitude save the equator are small circles on Earth. The flat planes that any of them lie upon cut across the Earth's polar axis at some distance north or south of the actual center. Nearly all circles of position that you plot as Sumner lines or LOP's in celestial are also small circles. Small circles do not share the handy characteristics of great circles, so be sure not to conflate the two.
For our purposes it is useful to alter our perspective a bit. We say that any circle on a sphere that has the same circumference as the maximum circumference of the sphere will lie in a flat plane that cuts the sphere exactly into two halves passing through the center of the sphere, and is therefore a great circle.
Imagine a schoolroom globe of the Earth. Now imagine you have made a bracelet that slips down over this globe so that it sits exactly on the equator. The plane it lies in is the equatorial plane of the globe which we know cuts the globe into two equal halves and passes through the exact three dimensional center of the globe. This bracelet is a great circle on the globe and it represents a great circle on the real Earth.
Next imagine that you can slip and slide this bracelet anywhere you want on your globe provided it stays in full contact with the globe. In other words if you slide the near side up the back side slides down, but the whole bracelet stays in contact with the globe. The circumference of the globe and the circumference of the bracelet stay the same regardless of where you slide the bracelet because the globe is a sphere. No matter where you slip the bracelet on the globe it still has a circumference equal to the maximum circumference of the globe and so it always remains a great circle on the globe – and it always represents a great circle on Earth.
Realizing that the circumference of the bracelet never changes you mark a scale along it in degrees from 0° to 360° and now you can slide the bracelet anywhere you want on the globe to connect any two points and measure the distance between them in degrees.
Because the real Earth is (almost) a sphere and all of Earth's great circles have the same circumference we adopted the convention that on Earth one degree along a great circle is 60 nautical miles. That convention won't work on other planets because they don't have the same circumference as Earth, but for the moment we are only concerned with navigation on Earth. This convention was not arrived at by accident or coincidence – it was deliberately contrived for the purposes of navigation. If we can measure the angle between any two points on Earth along a great circle with our globe and bracelet we can now easily convert that measurement into nmi. What follows explains why we take the measure in degrees instead of just directly measuring it in nautical miles.
Now consider the celestial sphere way out beyond the surface of the Earth. The celestial sphere shares the exact same center as Earth's center. Therefore any flat plane that passes through Earth's center and defines a great circle on Earth can be extended out to the celestial sphere where it forms another great circle out there. In our globe and bracelet model this means that no matter where we slide our bracelet it not only represents a great circle on Earth, it also represents a matched great circle out on the celestial sphere. These two great circles work as a matched pair.
Consider a classic wagon wheel. Whatever the angle is between any two spokes on the rim it will be the same angle between those spokes at the hub – because they share the same center.
In our model if we take two places on Earth, A and B, and find their zenith points on the celestial sphere, then the angle between those two zenith points when measured along a great circle on the celestial sphere will be the same angle as points A and B are apart when measured along the matched great circle on Earth, provided we make the measurement in degrees of arc and not in linear measurements such as nautical miles.
It turns out this is very handy because in celestial navigation we need to figure out how far we are standing from a star's geographic position (GP). Usually the GP is very far away from us and below our horizon, so we can't see it to measure it directly – but we can see our zenith and we can see the star sitting directly over its own GP. The angle measure from the zenith over to the star along a celestial sphere great circle is exactly the same as the angle measure from beneath our feet over to the star's GP along an Earth great circle – and the Earth version is easily converted into nmi. If we could reliably take the measurement from our zenith over to the star we would know how far we were standing from the GP of the star and our immediate problem would be solved.
It turns out that it is difficult to measure using your precise zenith as a reference, but it is relatively easy to use the horizon, and with a few simple corrections determine the star's height above the true horizontal. (Ho) Next we apply the knowledge that the entire angle from the true horizontal to our zenith is always 90° and so the distance from the zenith to the star (called the zenith distance) must be whatever is left over from 90° after you subtract Ho. Zenith distance = (90° – Ho). And this is the angular distance from you to the star's GP which, when converted into nmi, allows you to plot a circle of position centered around the star's GP at a known radius in nautical miles.
Actually for most sight reduction methods we never state the zenith distance explicitly, nevertheless it is there inside the “mathematical black box” of whatever sight reduction method you choose to use. Ho is related to the observed zenith distance. Calculated altitude from an an assumed position (Hc) is related to the zenith distance from the assumed position to which we will to compare our Ho. When we combine them to find the intercept the 90° parts from each zenith distance term cancel each other out.
(90° – Hc) – (90° – Ho) => (90° – 90°) + (Ho – Hc) => (Ho – Hc)
The last thing we need to do is to convince ourselves that we are measuring the distance up from the horizon to the star along an actual great circle on the celestial sphere.
When we take a shot of a star we face in the exact direction as the light rays coming from the star. Since the star is directly over its own GP that is the exact same direction as from us to the GP. Next we go to some pains to hold the sextant vertical as we take our measurement.
There are two light rays coming to our eye through the sextant. One is from the star and the other is from the horizon. We rock the sextant to find the spot where the ray from the horizon is straight down beneath the ray from the star and that is when we take our measurement. One way to define “straight down” is to say directly towards the center of the Earth. So these two light rays form a flat plane, and that plane can be extended straight down to go through the center of the Earth – and extended all the way out to the celestial sphere. Because such a plane forms great circles on both spheres we conclude that we are indeed measuring Ho along a great circle on the celestial sphere aligned with the direction of the star, and that it corresponds exactly with the great circle from beneath our feet over to the GP of the star.
I suppose you don't need to know all of this to practice celestial, and some people may find it a bit too esoteric while others may find my math too simple. But it does explain how it actually works. It was a great help to me when I finally figured it out and perhaps it will be a help to others. I know none of the texts I have explain it like this, which I hope is straightforward enough for most people to understand if they ponder it for a bit.
Other aspects that are germane to great circles also helped me a great deal when I studied the sailings – and that all began with the simple globe-and-bracelet mental model I've outlined above. But that is a discussion for another day.
PeterB
Many of us started out in celestial with the analogy of a flag pole and how the top of the flag pole appears at lower angles the further we stand from its base. This is a flat-Earth analogy that can only take you so far and sooner or later runs into limitations.
To begin with we must understand that all of the stars are so far away that the direction in which we see any stars doesn't change with our location on Earth.
For the moment consider Polaris, which seems to stand nearly still as the other stars rotate around it. If the entire Earth were the size of the period at the end of this sentence at that scale Polaris would be many times further away from the ink dot Earth than the Eiffel Tower is from New York City's Central Park. Clearly taking the bearing from one side of the ink dot in Central Park to the Eiffel Tower and then moving all the way to the complete opposite side of the ink dot and taking it again would not change the bearing any measurable amount.
In fact the real distance to Polaris is so far that even if you wait six months for the Earth to reach the opposite side of its entire orbit around the sun, the direction to Polaris won't change. What causes the change of the angle at which you see Polaris is a change in what you perceive as “horizontal” or “straight up” as you move about on the spherical Earth.
If we could transport ourselves far out into space and look back at Earth to see three people taking simultaneous observations of Polaris we would see that they are all looking in the exact same direction – at a star that appears to us to be directly over the Earth's north pole but which is incredibly far away. From our outer space location we would see that all three would be gazing northward in a direction parallel to Earth's polar axis. But Paul at the north pole sees the star directly over his head; Eddie at the equator sees it low on his horizon; and Matt in the mid latitudes sees it part way up his sky. What is different for each of them is what they consider to be “straight up” and “horizontal” depending on where they are standing on the sphere that is Earth.
This is true for all the other stars as well. It is only the spinning of the Earth that gives them the appearance of motion. If, from our outer space location, we took a look at several people taking simultaneous observations of Arcturus we would see that they too would all be looking in the exact same direction at that star, but the heights at which they each perceive the star in their local sky would differ by their locations on the sphere that is Earth.
Arcturus travels over the Big Island of Hawaii each day, so let us say we watch a bunch of people taking sights of Arcturus just as it is directly above Hawaii. To get a good mental picture of this imagine that you have a globe in your hands and hold it so that Hawaii is straight up. Now from your perspective all of the observers will be looking “straight up” at your ceiling and far beyond to see Arcturus regardless of where they are standing on the globe. Of course the further they are on the globe from Hawaii the more their personal sense of horizontal will be tipped over. If a few of them observe Arcturus at the same angle up in their sky we could say that they were all standing on a circle of position determined by the same distance from the star's geographic position. Alternately we could say that they are all standing on the same circle of position determined by an equal amount of horizontal tilting from the 0° tilt that occurs at the geographic position of the star.
Now on to great circles.
A great circle is a specific concept in spherical geometry. It is the circle that is formed on the surface of the sphere by a flat plane that passes through the exact center of the sphere. All great circles on any particular sphere have their centers at the center of the sphere. They all have the same diameter as the sphere, and they all have the same circumference, which is the maximum circumference of the sphere.
Think of cutting an orange exactly in half with a sharp straight knife. The outer edge of the peel on each of the two halves is now a circle. If the cut went through the exact center of the orange this circle would be a great circle on the orange. If instead that sphere were a beach ball it is a great circle on the beach ball; If the sphere were a classroom globe that circle would be a great circle on that globe; If the sphere were the Earth it is a great circle on Earth.
Any plane that does not go through the exact center of the sphere still makes a circle where it intersects the surface, but that is not a great circle. Instead it is technically a “small circle” even if it is enormous in size. All lines of latitude save the equator are small circles on Earth. The flat planes that any of them lie upon cut across the Earth's polar axis at some distance north or south of the actual center. Nearly all circles of position that you plot as Sumner lines or LOP's in celestial are also small circles. Small circles do not share the handy characteristics of great circles, so be sure not to conflate the two.
For our purposes it is useful to alter our perspective a bit. We say that any circle on a sphere that has the same circumference as the maximum circumference of the sphere will lie in a flat plane that cuts the sphere exactly into two halves passing through the center of the sphere, and is therefore a great circle.
Imagine a schoolroom globe of the Earth. Now imagine you have made a bracelet that slips down over this globe so that it sits exactly on the equator. The plane it lies in is the equatorial plane of the globe which we know cuts the globe into two equal halves and passes through the exact three dimensional center of the globe. This bracelet is a great circle on the globe and it represents a great circle on the real Earth.
Next imagine that you can slip and slide this bracelet anywhere you want on your globe provided it stays in full contact with the globe. In other words if you slide the near side up the back side slides down, but the whole bracelet stays in contact with the globe. The circumference of the globe and the circumference of the bracelet stay the same regardless of where you slide the bracelet because the globe is a sphere. No matter where you slip the bracelet on the globe it still has a circumference equal to the maximum circumference of the globe and so it always remains a great circle on the globe – and it always represents a great circle on Earth.
Realizing that the circumference of the bracelet never changes you mark a scale along it in degrees from 0° to 360° and now you can slide the bracelet anywhere you want on the globe to connect any two points and measure the distance between them in degrees.
Because the real Earth is (almost) a sphere and all of Earth's great circles have the same circumference we adopted the convention that on Earth one degree along a great circle is 60 nautical miles. That convention won't work on other planets because they don't have the same circumference as Earth, but for the moment we are only concerned with navigation on Earth. This convention was not arrived at by accident or coincidence – it was deliberately contrived for the purposes of navigation. If we can measure the angle between any two points on Earth along a great circle with our globe and bracelet we can now easily convert that measurement into nmi. What follows explains why we take the measure in degrees instead of just directly measuring it in nautical miles.
Now consider the celestial sphere way out beyond the surface of the Earth. The celestial sphere shares the exact same center as Earth's center. Therefore any flat plane that passes through Earth's center and defines a great circle on Earth can be extended out to the celestial sphere where it forms another great circle out there. In our globe and bracelet model this means that no matter where we slide our bracelet it not only represents a great circle on Earth, it also represents a matched great circle out on the celestial sphere. These two great circles work as a matched pair.
Consider a classic wagon wheel. Whatever the angle is between any two spokes on the rim it will be the same angle between those spokes at the hub – because they share the same center.
In our model if we take two places on Earth, A and B, and find their zenith points on the celestial sphere, then the angle between those two zenith points when measured along a great circle on the celestial sphere will be the same angle as points A and B are apart when measured along the matched great circle on Earth, provided we make the measurement in degrees of arc and not in linear measurements such as nautical miles.
It turns out this is very handy because in celestial navigation we need to figure out how far we are standing from a star's geographic position (GP). Usually the GP is very far away from us and below our horizon, so we can't see it to measure it directly – but we can see our zenith and we can see the star sitting directly over its own GP. The angle measure from the zenith over to the star along a celestial sphere great circle is exactly the same as the angle measure from beneath our feet over to the star's GP along an Earth great circle – and the Earth version is easily converted into nmi. If we could reliably take the measurement from our zenith over to the star we would know how far we were standing from the GP of the star and our immediate problem would be solved.
It turns out that it is difficult to measure using your precise zenith as a reference, but it is relatively easy to use the horizon, and with a few simple corrections determine the star's height above the true horizontal. (Ho) Next we apply the knowledge that the entire angle from the true horizontal to our zenith is always 90° and so the distance from the zenith to the star (called the zenith distance) must be whatever is left over from 90° after you subtract Ho. Zenith distance = (90° – Ho). And this is the angular distance from you to the star's GP which, when converted into nmi, allows you to plot a circle of position centered around the star's GP at a known radius in nautical miles.
Actually for most sight reduction methods we never state the zenith distance explicitly, nevertheless it is there inside the “mathematical black box” of whatever sight reduction method you choose to use. Ho is related to the observed zenith distance. Calculated altitude from an an assumed position (Hc) is related to the zenith distance from the assumed position to which we will to compare our Ho. When we combine them to find the intercept the 90° parts from each zenith distance term cancel each other out.
(90° – Hc) – (90° – Ho) => (90° – 90°) + (Ho – Hc) => (Ho – Hc)
The last thing we need to do is to convince ourselves that we are measuring the distance up from the horizon to the star along an actual great circle on the celestial sphere.
When we take a shot of a star we face in the exact direction as the light rays coming from the star. Since the star is directly over its own GP that is the exact same direction as from us to the GP. Next we go to some pains to hold the sextant vertical as we take our measurement.
There are two light rays coming to our eye through the sextant. One is from the star and the other is from the horizon. We rock the sextant to find the spot where the ray from the horizon is straight down beneath the ray from the star and that is when we take our measurement. One way to define “straight down” is to say directly towards the center of the Earth. So these two light rays form a flat plane, and that plane can be extended straight down to go through the center of the Earth – and extended all the way out to the celestial sphere. Because such a plane forms great circles on both spheres we conclude that we are indeed measuring Ho along a great circle on the celestial sphere aligned with the direction of the star, and that it corresponds exactly with the great circle from beneath our feet over to the GP of the star.
I suppose you don't need to know all of this to practice celestial, and some people may find it a bit too esoteric while others may find my math too simple. But it does explain how it actually works. It was a great help to me when I finally figured it out and perhaps it will be a help to others. I know none of the texts I have explain it like this, which I hope is straightforward enough for most people to understand if they ponder it for a bit.
Other aspects that are germane to great circles also helped me a great deal when I studied the sailings – and that all began with the simple globe-and-bracelet mental model I've outlined above. But that is a discussion for another day.
PeterB