Some Important Geometry. (Astronomy)

Great Circles:

The shortest path between two points on a plane is a straight line. On the surface of a sphere, however, there are no straight lines. The shortest path between two points on the surface of a sphere is given by the arc of the great circle passing through the two points. A great circle is defined to be the intersection with a sphere of a plane containing the center of the sphere.

Two great circles

If the plane does not contain the center of the sphere, its intersection with the sphere is known as a small circle. In more everyday language, if we take an apple, assume it is a sphere, and cut it in half, we slice through a great circle. If we make a mistake, miss the center and hence cut the apple into two unequal parts, we will have sliced through a small circle.

Two small circles


Spherical Triangles:

If we wish to connect three points on a plane using the shortest possible route, we would draw straight lines and hence create a triangle. For a sphere, the shortest distance between two points is a great circle. By analogy, if we wish to connect three points on the surface of a sphere using the shortest possible route, we would draw arcs of great circles and hence create a spherical triangle. To avoid ambiguities, a triangle drawn on the surface of a sphere is only a spherical triangle if it has all of the following properties:

  • The three sides are all arcs of great circles.
  • Any two sides are together greater than the third side.
  • The sum of the three angles is greater than 180°.
  • Each spherical angle is less than 180°.

Hence, in figure below, triangle PAB is not a spherical triangle (as the side AB is an arc of a small circle), but triangle PCD is a spherical triangle (as the side CD is an arc of a great circle). You can see that the above definition of a spherical triangle also rules out the “triangle” PCED as a spherical triangle, as the vertex angle P is greater than 180° and the sum of the sides PC and PD is less than CED.

The figure below shows a spherical triangle, formed by three intersecting great circles, with arcs of length (a,b,c) and vertex angles of (A,B,C).

Note that the angle between two sides of a spherical triangle is defined as the angle between the tangents to the two great circle arcs, as shown in the figure below for vertex angle B.


Earth’s Surface:

The rotation of the Earth on its axis presents us with an obvious means of defining a coordinate system for the surface of the Earth. The two points where the rotation axis meets the surface of the Earth are known as the north pole and the south pole and the great circle perpendicular to the rotation axis and lying half-way between the poles is known as the equator. Great circles which pass through the two poles are known as meridians and small circles which lie parallel to the equator are known as parallels or latitude lines.

The latitude of a point is the angular distance north or south of the equator, measured along the meridian passing through the point. A related term is the co-latitude, which is defined as the angular distance between a point and the closest pole as measured along the meridian passing through the point. In other words, co-latitude = 90° – latitude.

Distance on the Earth’s surface is usually measured in nautical miles, where one nautical mile is defined as the distance subtending an angle of one minute of arc at the Earth’s center. A speed of one nautical mile per hour is known as one knot and is the unit in which the speed of a boat or an aircraft is usually measured.


Horizon System:

Humans perceive in Euclidean space -> straight lines and planes. But, when distances are not visible (i.e. very large) than the apparent shape that the mind draws is a sphere -> thus, we use a spherical coordinate system for mapping the sky with the additional advantage that we can project Earth reference points (i.e. North Pole, South Pole, equator) onto the sky. Note: the sky is not really a sphere!

From the Earth’s surface we envision a hemisphere and mark the compass points on the horizon. The circle that passes through the south point, north point and the point directly over head (zenith) is called the meridian.

This system allows one to indicate any position in the sky by two reference points, the time from the meridian and the angle from the horizon. Of course, since the Earth rotates, your coordinates will change after a few minutes.

The horizontal coordinate system (commonly referred to as the alt-az system) is the simplest coordinate system as it is based on the observer’s horizon. The celestial hemisphere viewed by an observer on the Earth is shown in the figure below. The great circle through the zenith Z and the north celestial pole P cuts the horizon NESYW at the north point (N) and the south point (S). The great circle WZE at right angles to the great circle NPZS cuts the horizon at the west point (W) and the east point (E). The arcs ZN, ZW, ZY, etc, are known as verticals.

The two numbers which specify the position of a star, X, in this system are the azimuth, A, and the altitude, a. The altitude of X is the angle measured along the vertical circle through X from the horizon at Y to X. It is measured in degrees. An often-used alternative to altitude is the zenith distance, z, of X, indicated by ZX. Clearly, z = 90 – a. Azimuth may be defined in a number of ways. For the purposes of this course, azimuth will be defined as the angle between the vertical through the north point and the vertical through the star at X, measured eastwards from the north point along the horizon from 0 to 360°. This definition applies to observers in both the northern and the southern hemispheres.

It is often useful to know how high a star is above the horizon and in what direction it can be found – this is the main advantage of the alt-az system. The main disadvantage of the alt-az system is that it is a local coordinate system – i.e. two observers at different points on the Earth’s surface will measure different altitudes and azimuths for the same star at the same time. In addition, an observer will find that the star’s alt-az coordinates changes with time as the celestial sphere appears to rotate. Despite these problems, most modern research telescopes use alt-az mounts, as shown in the figure above, owing to their lower cost and greater stability. This means that computer control systems which can transform alt-az coordinates to equatorial coordinates are required.

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