Geometry for parallax measurements.The figure is not to scale. In reality the distance to the star, d, is much greater than 1 AU, so the parallax angle, p, would normally be very small.
where p(") is the parallax angle measured in arc seconds. Substituting this into equation (2.15) gives
The natural unit for measuring angles is the radian. If we have a circle of radius R, and two lines from the center making an angle 9 with each other then the length of the arc bounded by the two lines is
where 9(rad) is the value of 9 measured in radians. Since the full circumference of a circle is 2wR, the angle corresponding to a full circle must be 2w radians.This tells us that a full circle, that is 360°, is equal to 2w radians, or 180° is equal to w radians. In astronomy, we often deal with very small angles, and measurements in arc seconds (") are convenient. We can convert measurements by saying
When we take the derivatives of trigonometric functions (for example, d(sin 9)/d9 = cos 9), it is assumed that the angles are in radians. If not, a conversion factor must be carried through the differentiation.
When angles expressed in radians have a value that is much less than unity we can use a Taylor series to approximate them:
This gives the distance to the star in AU (1 AU = 1.50 X 108 km).
This method suggests a convenient unit for measuring distances. We define the parsec (abbreviated pc) as the distance of a star that produces a parallax angle p of 1 arc sec. From equation (2.16), we can see that 1 pc = 2.06 X 105 AU (or 3.09 X 1013 km, or 3.26 light years). We rewrite equation (2.16) as d(pc) = 1/p(")
Remember, as an object moves farther away, the parallax angle decreases. Therefore, a star at a distance of 2 pc will have a parallax angle of 0.5 arc sec.
Note that for small 9, sin 9 and tan 9 are both approximately equal to 9, so they must be equal to each other (Remember in each of the above expressions, 9 must be expressed in radians.)
Example 2.3 Distance to the nearest star The nearest star (Proxima Centauri) has a parallax p = 0.76 arc sec. Find its distance from Earth in parsecs.
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