The Apparent Orbit

The apparent (observed) orbit results from a projection of the true orbit onto the celestial sphere (Figure 7.4). Three more elements determine this projection:

Q the position angle of the ascending node. This is the position angle of the line of intersection between the plane of projection and the true orbital plane. The angle is counted from north to the line of nodes. The ascending node is the node where the motion of the companion is directed away from the Sun. It differs from the second node by 180° and can be determined only by radial-velocity measurements. If the ascending node is unknown, the value < 180° is given.

Ascending Node

Figure 7.4. The projected elements of a visual binary star.

Figure 7.4. The projected elements of a visual binary star.

i the orbital inclination. This is the angle between the plane of projection and the true orbital plane. Values range from 0° to 180°. For 0° < i < 90° the motion is called direct. The companion then moves in the direction of increasing position angles (anticlockwise). For 90° < i < 180° the motion is called retrograde. ra the argument of periastron. This is the angle between the node and the periastron, measured in the plane of the true orbit and in the direction of the motion of the companion.

The elements P, T, a, e, i, ra, Q, are called the Campbell elements. There is another group of elements which is used in order to calculate rectangular coordinates. They are called Thiele-Innes elements (Figure 7.5):

A = a (cos ra cos Q - sin ra sin Q cos i) B = a (cos ra sin Q + sin ra cos Q cos i)

F = a (-sin ra cos Q - cos ra sin Q cos i) G = a (-sin ra sin Q + cos ra cos Q cos i).

node = 107 A = -0.060 B = -0.208 F = +0.181 G = -0.438 e = 0.80 $ = 53.13

node = 107 A = -0.060 B = -0.208 F = +0.181 G = -0.438 e = 0.80 $ = 53.13

auxiliary circle auxiliary circle

North

Note that the elements A, B, F and G are independent of the eccentricity e. The points (A, B) and (F cos G cos together with the centre of the apparent ellipse, define a pair of conjugate axes which are the projections of the major and minor axes of the true orbit.

There is an instructive and easy way to draw the apparent orbit from the seven Campbell elements. It runs as follows:

1. Draw the rectangular coordinate system with a convenient scale. North is at the bottom (the positive x-axis); east is at the right (the positive y-axis).

2. Draw the line of nodes. The node makes the angle Q between north and the line of nodes.

3. Lay off the angle ra from the line of nodes and proceeding in the direction of the companion's motion, i.e. clockwise, when i > 90°, and counterclockwise,

Figure 7.5.

Thiele-Innes elements and Campbell elements.

Figure 7.6. The true and the projected orbit of OS 235 drawn in one plane. Note: the law of areas holds in the projected ellipse as well.

Figure 7.6. The true and the projected orbit of OS 235 drawn in one plane. Note: the law of areas holds in the projected ellipse as well.

Periastron

when i < 90°. This will give the line of periastron and apastron of the true orbit.

4. Draw the true orbit ellipse. The distance of the centre of the true orbit from the centre of the coordinate system is c. The long axis is 2a, the short axis is 2b, so b and c are easily calculated:

5. Construct the apparent orbit. Draw lines from points on the true orbit to the line of nodes; the lines have to be perpendicular to the line of nodes. Multiply the lines by cos i. Connecting the so obtained points yields the apparent orbit.

As an example, the orbit for OX 235 is given in Figure 7.6. Elements are as follows (Heintz1): P = 73.03 years, T = 1981.69, a = 0'.'813, e = 0.397, i = 47°3, ra = 130°9, Q = 80°9.

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