The values of 0 are so small that radians are an inconvenient quantity. We can convert to arc seconds (equation (2.16)) to give

The factor of 2.06 X 105 was to convert radians to arc seconds, but it is also the factor to convert astronomical units to parsecs, so we have

If we use equation (2.17) to relate the distance in parsecs to the parallax in arc seconds, this becomes

For the Solar System, we write the sum of the masses as one in these units, so the equation simply says that the cube of the radius (in AU) is equal to the square of the period (in yr). This is also known as Kepler's third law of planetary motion. The law was originally found by Kepler observa-tionally, and Newton used it to show that gravity must be an inverse square law force. (See Problem 5.10.) We will discuss planetary motions in more detail in Chapter 22.

For a visual binary, we don't directly measure R, the linear separation. We actually measure the angular separation on the sky, 0. If d is the distance to the binary, then R is equal to 0(rad)d, where 0(rad) is the value of 0 measured in radians. When we use this relation, R and d will come out in the same units. Therefore

This can then be substituted directly into equation (5.21).

We will now look at the behavior of the Doppler shifts. Applying equation (5.10) to both speeds, v1 and v2, and remembering that the period of the orbit is the same for both stars, we have r1 + r2 = (P/2w )(v1 + v2)

Using this to eliminate R in equation (5.20) gives

If the orbit is inclined at an angle i, then the Doppler shifts only measure the components vr = vsin(i). In terms of the radial velocities v1r and v2r, equation (5.23) becomes

If a binary happens to be an eclipsing binary, then we know that we are close to the plane of the orbit, and i is close to 90°. Otherwise we don't know i. If a circular orbit is projected at some angle on the plane of the sky it will appear elliptical. We will see in the next section that there are ways to determine i if we can trace that projected orbit on the sky. If i is unknown all we can do is solve equation (5.24) with i = 90°. This will give us a value of m1 + m2 that is a lower limit to the true value. The true value would be this lower limit divided by sin3i, and since sin3i is less than or equal to unity, the value assuming i = 90° is less than the true value. Finding lower limits is not as useful as finding actual values. However, if we study enough binary systems, we will encounter a full range of inclination angles.

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