The Doppler Method for the Detection of Exoplanets. Professor Artie Hatzes

is easy to calculate how the RV changes if the same detector were moved to a spectrograph with higher resolving power, i.e., for a “fixed-size detector.” In this case, what you gain in precision from the increased resolving power is partially offset by the loss in precision due to the smaller wavelength coverage. The uncertainty due to resolving power scales as σR∝R−α. The uncertainty with wavelength coverage, Δλ, scales as σΔλ ∝(Δλ)−1/2. The wavelength coverage scales as R−1, so substituting into the previous expression, one gets σΔλ∝R1/2. So, for the case of the fixed-sized detector, the total uncertainty is given by the product of the two, namely σTotal∝R1/2−α.

      Figure 3.5 shows the actual RV error determined from solar spectra (day sky observations) using an iodine absorption cell at resolving powers of R = 2300, 15,000, and 200,000. These were taken with the same detector and spectrograph, but different gratings to provide different resolutions. The solid red curve is the function σ∝R−1, while the dotted curve is σ∝R−1/2. At first glance this seems to support σ∝R−1. Therefore, (1/2−α)=−1, which implies that α = 3/2 as opposed to unity. Keep in mind the caveat that these data were taken with the iodine absorption cell.

      Figure 3.5. (Points) The radial velocity error taken with a spectrograph at different resolving powers. This is the actual data taken of the day sky all with the same S/N values. The solid red line shows a σ∝R−1 fit. The dashed black line shows a σ∝R−1/2 fit. The detector size is fixed for all data, thus the wavelength coverage is increasing with decreasing resolving power.

3.2 Stellar Characteristics

      The properties of the star also have a large influence on your RV precision. The RV uncertainty depends on three major fundamental features of the star (stellar spectrum):

       The projected rotational velocity of the star, or sin i.

       The strength of stellar spectral lines.

       The number density of stellar lines.

3.2.1 Stellar Rotational Velocity

      In the absence of stellar variability, stellar rotation has the largest influence on the RV measurement error. Rotation broadens the width of the stellar lines and makes them shallower, thus making it more difficult to determine the centroid. Figure 3.6 shows the spectral region of two stars, the top of a B9 star and the lower panel of a K5 star. The hot star only has one spectral line in this region, and it is quite broad and shallow, due to the high projected rotation rate of the star, sin i, where is the true rotational velocity of the star and i is the inclination of the spin axis to the observer. Clearly, it would be more difficult to determine the centroid position and thus a Doppler shift of the rotationally broadened spectral line.

      Figure 3.6. (Top) A spectrum of a B9 star rotating at 230 km s⁻¹. (Bottom) The spectrum of a K5 star.

      Table 3.2 lists the median sin i as a function of stellar types for main-sequence stars taken from the Glebocki et al. (2000). Note the sharp drop in rotational velocities at mid-F, which is often called the “rotation break” or “Kraft break,” which was first noted by Kraft (1967). This results from the fact that around mid-F, stars start to develop a substantial outer convection zone. This, coupled with rotation, leads to magnetic activity, and the star loses angular momentum through magnetic braking.

Table 3.2. Median Rotational Velocities of Stars

Spectral Type	sin i (km s−1)
O4	110
O9	105
B5	108
A0	82
A5	80
F0	44
F5	11
G0	4
G5	3
K0	3
K5	2
M0	10
M4	16
M9	10

      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 sin i and different resolving powers of the spectrograph. These curves were generated by measuring the relative shift of a single synthetic spectral line at the appropriate resolution and sin i. Noise at a level of S/N = 50 was added, but the shape of the curves are the same for a fixed S/N. For each curve (fixed R), the uncertainty was normalized to the value of sin i ≈2 km s⁻¹. The ordinate thus represents the factor by which the uncertainty scales with the stellar rotational velocity.

      Figure 3.7. The scale factor for the increase in the RV uncertainty as a function of the stellar sin i and for several values of the resolving power (R = 15,000–200,000). Each curve has been normalized to the uncertainty for sin i = 2 km s⁻¹.

      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 sin i by scaling the known performance of the same spectrograph with a given resolving power. As an example, suppose you use a spectrograph