S-Index and RHK

class PyAstronomy.pyasl.SMW_RHK(ccfs='rutten', afc='middelkoop', rphot='noyes')

Converting Mount-Wilson S-index into RHK index.

The Mount-Wilson S-index is a measure of the flux in the emission-line cores of the Ca II H and K lines at about 3933 A and 3968 A in two narrow bands normalized by the flux in two adjacent continuum bands (3891.067-3911.069 A and 3991.067-4011.067 A). For technical reasons, the narrow bands have a triangular bandpass with a FWHM of 1.09 A (Vaughan et al. 1978).

The activity index RHK is closely related to the S-index. In particular, it gives the emission in the narrow bands normalized by the bolometric brightness of the star

\[R_{HK} = \frac{4\pi R_s^2 (F_H + F_K)}{4\pi R_s^2\sigma T_{eff}^4} = \frac{F_H+F_K}{\sigma T_{eff}^4} \; .\]

The stellar surface flux in “arbitrary units” in the narrow H and K bands (fH + fK) is related to the Mount-Wilson S-index through the relation

\[f_H + f_K = S C_{cf} T_{eff}^4 10^{-14} \; ,\]

where Ccf is a conversion factor, which can be parameterized in terms of the B-V color and the luminosity class. The conversion between arbitrary units and physical units, needed to derive the true surface flux and the RHK index, has been derived by several authors starting with Middelkoop 1982. Their factor was also used by Noyes et al. 1984—in particular in their appendix a, where is appears implicitly. Later, the value of the conversion factor has been revised by several authors, e.g., Oranje 1983 and Rutten 1984, who estimated a value about 70% larger than previously proposed. Hall et al. 2007 derive a value 40% larger than that of Middelkoop 1982 and provide a thorough discussion on the differences between the individual approaches and results given in the literature.

The RHK index thus derived still contains a contribution of photospheric flux in the narrow bands, which is always present and not related to the chromosphere. To obtain the purely chromospheric, primed RHK index, an estimate of the photospheric surface flux in the H and K pass-bands has to be subtracted. For active stars, the photospheric correction is usually quite irrelevant. For inactive, quiet stars, it can, however, be important.

The issue of the Mount-Wilson S-index conversion has been revisited by Mittag et al. 2013, who provide an alternative conversion procedure and revised photospheric corrections for various luminosity classes.

Note

In the default configuration, the conversion of the S-index into RHK is identical to the relation stated by Noyes et al. 1984 in their appendix (a)

\[R_{HK} = 1.340 \times 10^{-4} C_{cf} S\]

where the factor 1.34e-4 is a combination of the conversion from arbitrary to physical units, 1e-14, and the Stefan-Boltzmann constant, in particular 1.34e-4 = 7.6e5*1e-14/5.67e-5. The Ccf factor is, however, calculated according to Rutten 1984.

The relations and coefficients used here are taken from the following publications (and references therein):

  • Middelkoop 1982, A&A 107, 31

  • Oranje 1983, A&A 124, 43

  • Noyes et al. 1984, A&A 279, 763

  • Rutten 1984, A&A 130, 353

  • Hall et al. 2007, AJ 133, 862

  • Mittag et al. 2013, A&A 549, 117

  • Vaughan et al. 1978, PASP 90, 267

Parameters
ccfsstring, {rutten, noyes}, optional

Source of the conversion factor between S-index and RHK.

afcstring, {rutten, oranje, middelkoop, hall}, optional

Source of conversion factor between “arbitrary units” and physical units of surface flux.

rphotstring, {noyes}

The source for the photospheric correction for the RHK index.

Methods

FHFK(S, Teff, log10ccf)

Calculate the FH+FK flux in arbitrary units.

SMWtoRHK(S, Teff, bv[, lc, verbose])

Convert Mount-Wilson S-index into R_HK.

log10ccfNoyes(bv, **kwargs)

Ccf conversion factor according to Noyes et al. 1984.

log10ccfRutten(bv[, lc])

Ccf conversion factor from Rutten 1984 (Eqs.

logRphotNoyes(bv[, lc])

Photospheric contribution to surface flux in the H and K pass-bands.
FHFK(S, Teff, log10ccf)

Calculate the FH+FK flux in arbitrary units.

Parameters
Sfloat

Mount-Wilson S-index.

Tefffloat

The effective temperature [K].

log10ccffloat

The logarithm of the Ccf conversion factor.

Returns
FH + FKfloat

The stellar surface flux in the H and K pass-bands in arbitrary units (not erg/cm**2/s).

SMWtoRHK(S, Teff, bv, lc='ms', verbose=False)

Convert Mount-Wilson S-index into R_HK.

Parameters
Sfloat

Mount-Wilson S-index.

Tefffloat

Effective temperature [K].

bvfloat

B-V color [mag]

lcString, {ms, g}, optional

Luminosity class; Main-sequence (ms) or giants (g)

verboseboolean, optional

If True, the details of the calculation are printed to stdout.

Returns
RHK primefloat

RHK parameter corrected for photospheric contribution. The primed number measures the purely chromospheric emission.

RHKfloat

RHK parameter without correction for photospheric contribution.

ccffloat

The Ccf conversion factor used.

fhfkfloat

The FH+FK surface flux in arbitrary units.

fhfk (physical)float

The FH+FK surface flux in physical units [erg/cm^2/s].

R_photfloat

Photospheric flux contribution used in translating RHK into RHK prime.

log10ccfNoyes(bv, **kwargs)

Ccf conversion factor according to Noyes et al. 1984.

Parameters
bvfloat

The B-V color [mag].

Returns
log10(Ccf)float

The logarithm of the conversion factor.

log10ccfRutten(bv, lc='ms')

Ccf conversion factor from Rutten 1984 (Eqs. 10a and 10b).

Parameters
bvfloat

B - V color [mag].

lcstring, {ms, g}, optional

Specifies whether the relation for main-sequence (ms) or giant (g) stars shall be evaluated.

Returns
log10(Ccf)float

The logarithm of the conversion factor.

logRphotNoyes(bv, lc='ms')

Photospheric contribution to surface flux in the H and K pass-bands.

Relation given by Noyes et al. 1984.

Parameters
bvfloat

B-V color [mag]

lcstring, {ms, g}, optional

Luminosity class.

Returns
log10(Rphot)float

Logarithm of the photospheric contribution.

Convert Mount-Wilson S-index into RHK

from __future__ import print_function, division
from PyAstronomy import pyasl

ss = pyasl.SMW_RHK()

bv = 0.8
teff = 5100.0
s = 0.4

print("Convert S-index to RHK assuming a giant")
ss.SMWtoRHK(s, teff, bv, lc="g", verbose=True)

print()
print()
print("Convert S-index to RHK assuming a main-sequence star")
ss.SMWtoRHK(s, teff, bv, lc="ms", verbose=True)

Show the Ccf conversion factor

from PyAstronomy import pyasl
import numpy as np
import matplotlib.pylab as plt

ss = pyasl.SMW_RHK()

bv = np.arange(0.4, 0.9, 0.05)
ccfn = bv * 0.0
ccfr = bv * 0.0
ccfrg = bv * 0.0

for i in range(len(bv)):
    ccfn[i] = ss.log10ccfNoyes(bv[i])
    ccfr[i] = ss.log10ccfRutten(bv[i])
    ccfrg[i] = ss.log10ccfRutten(bv[i], lc="g")

plt.plot(bv, ccfn, 'b.-', label="Noyes")
plt.plot(bv, ccfr, 'r.-', label="Rutten (ms)")
plt.plot(bv, ccfrg, 'g.-', label="Rutten (g)")
plt.xlabel("B - V [mag]")
plt.ylabel("Ccf")
plt.legend()
plt.show()