In terms of good RV precision, early-type stars are poor targets for RV measurements for two reasons. First, these stars are hot, and as such, they have much fewer spectral lines for RV measurements than for stars at the lower end of the main sequence. This is seen in the lower panel of Figure 3.6, where the K5 star has a higher density of spectral lines. Second, rapid rotation greatly degrades the RV precision.
Figure 3.7 shows the behavior of the RV uncertainty as a function of
Figure 3.7. The scale factor for the increase in the RV uncertainty as a function of the stellar
Equation (3.3) and the curves in Figure 3.7 should only be used to estimate the RV uncertainty for a star with a certain
Formally, the curves can be fit by the function
f(V)∝0.62+(0.21logR−0.86)V+(0.00260logR−0.0103)V2,(3.3)
where V is the projected rotational velocity in km s−1.
One can also use more simple relationships to scale between stars of different rotational velocities depending on whether you have slow or fast rotating stars.
For typical resolving powers of spectrographs used for precise RV measurements, the RV precision due to stellar rotation scales as:
for
for
For example, if you have an RV precision of 3 m s−1 on a star rotating at 2 km s−1, then you should get an RV precision of approximately 4 m s−1 on a star rotating at 5 km s−1. Likewise, a star rotating at 70 km s−1 will have an RV a factor of 2 larger than a star rotating at 40 km s−1.
Note that the latter equation has a slightly higher dependence on
3.2.2 Spectral Line Strength
The RV precision depends on the depth of the stellar line. Clearly, if your line is too weak it will get lost in the noise, and an RV measurement will be next to impossible. Let us define the depth, d, as a value from 0 to 1. A line with d = 1.0 has a depth that is 100% of the continuum, i.e., zero flux in the core of the line. A weak line with d<1.0 will produce a measurement error that is some factor, F, times the measurement error of the stronger line.
Figure 3.8 shows a simulation of the Doppler measurement error as a function of spectral lines of fixed width, but varying depths. This simulation used an S/N = 50, but the results are insensitive to the exact S/N chosen. It shows that 1/F scales linearly with line depth. That is to say, if a line has a depth of one-fifth the continuum, it will have an RV measurement error that is five times greater than that of a line with a depth of 100% of the continuum.
Figure 3.8. The inverse of the multiplicative factor in the RV uncertainty, F, as a function of line strength. (Blue triangles) The factor in the uncertainty for spectral lines of constant width, but a depth that is a fraction of the continuum value (top abscissa). (Red squares) The factor in the RV uncertainty as a function of equivalent width (lower abscissa) for a real spectral line. The best RV precision is for strong, unsaturated lines.
However, real spectral lines do not have a depth that scales linearly with the line strength. Rather, these follow the so-called curve of growth. As one increases the line strength as measured by the equivalent width (EW), the line depth increases, but the width remains fairly constant (Figure 3.9). Once the line starts to saturate, the line depth remains constant, but the wings, and thus the line width, starts to increase. The triangles in Figure 3.8 show how the factor F varies with the EW for real spectral lines. At first, the curve follows the one for lines of fixed width, but after EW≈100 mÅ, the curve starts to flatten out. For the strongest lines, the RV uncertainty actually starts to increase with increasing line strength (Figure 3.8).
Figure 3.9. The change in spectral line shape as a function of increasing line strength.
This behavior can easily be understood in terms of how the RV uncertainty varies with line depth and widths. For weak spectral lines, the increase in EW is due primarily to an increase in the line depth, so the RV uncertainty follows the behavior in our simple simulation (squares in Figure 3.8). Once the line saturates,
