5.5.1 Stellar activity and rotation

It is well known that stellar chromospheric activity, as measured by the strength of emissions lines such as Ca II HK (3934 A), Mg II (2800 A), C iv (1550 A), N v (1240 A) and Lya (1216 A), decreases with main-sequence age, t. Magnetic activity, which controls chromospheric, transition region and coronal activity, is driven by the interaction of stellar rotation, internal differential rotation and convection (the dynamo action). The primary indicator of MS activity is thus the stellar rotation rate, which varies with MS age essentially as t-1/2. In addition, the depth of a star's convection zone is thought to play a role in the dynamo regeneration of magnetic fields. The deeper the convection zone the larger are the

Star |
Spectral type |
Age (Byr) |
P(d) |
Age indicator |

EK Dra |
GO V |
0.1 |
2.68 |
Local Association and Li |

n1 UMa |
G1.5 V |
0.30 |
4.90 |
UMa group |

X1 Ori |
G1 V |
0.30 |
5.24 |
UMa group |

9 Cet |
G3 V |
0.65 |
T.6 |
Hyades stream |

K1 Cet |
G5 V |
0.65 |
9.21 |
Prot-age & Lx |

ß Com |
G0 V |
1.6 |
12.0 |
Prot-age rel. |

15 Sge |
G1 V |
1.9 |
13.5 |
Prot-age rel. |

Sun |
G2 V |
4.6 |
25.4 |
Isotopic dating |

18 Sco |
G2 V |
4.9 |
23.0 |
Isochrones |

ß Hyi |
G2 IV |
6.T |
-28 |
Isochrones |

16 Cyg A |
G1.5 V |
8.5 |
-35 |
Isochrones |

convection cells and hence what is called the convective overturn time, tc. Thus both the stellar rotation period, P (days), and depth of the convection zone determine stellar chromospheric activity, and hence ultraviolet flux. Stars less massive than the Sun rotate more slowly at a given MS age, while higher-mass stars rotate faster. The cooler MS stars (increasing B-V) have deeper convection zones so that the convective overturn time increases with B-V. Further, the higher the metallicity (proportion of elements heavier than H), the deeper is the convection zone due to increased opacity by these elements. Chromospheric and transition-region activity, which determines radiation emission from Ca HK to Lya and XUV, has been correlated with the Rossby number, K. This parameter is a measure of the rate of fluid flow (convection speed) against rotational speed and is given by

We shall give simple expressions to derive K also from spectral type (B-V) and age. The stellar Rossby number and effective temperature (Teff) can be used to calculate stellar surface XUV and Lya emission flux from G-type solar-like stars, relative to present solar values. Furthermore, given the spectral type, effective temperature and radius with age, one can calculate the variation of the ultraviolet flux emission from such stars and hence, through photochemical models (see Chapter 7), its effects on the evolution of atmospheres of planets orbiting such stars. The above are also directly applicable to solar ultraviolet flux evolution, and hence to the evolution of the Earth's atmosphere from the Precambrian era (see Chapter 12). We shall give the evolution of the solar XUV and Lya emission flux with age from 0.1 to 8.5 Byr, based on the model of Girardi et al. (2000), for a 1 solar mass star with solar metallicity.

For solar-like stars, Ayres (1997) found a rotation period-age relation P x t0 6, while Barry (1988) found for G stars a spin-down dependence on spectral type according to P x e-A, where the parameter A is a function of B-V. Here, we shall combine the rotation-age relation with the rotation period-spectral type dependence to obtain the age evolution of the rotational period for each spectral type according to

where Pq is the present solar rotational period of 26.09 days (Messina et al. 2003) and the MS age t is measured in Byr. The above expression has been normalized to the present solar rotation period with a present solar age tQ = 4.56 Byr, with Aq = 0.77 corresponding to (B — V)q = 0.65 based on the work of Friel (1993). The problem of the solar B-V has been discussed by Taylor (1994) who proposed a value of 0.63, while the solar evolution models of Girardi et al. (2000) give a value of 0.67 for the present Sun. The parameter A is tabulated by Barry (1988) but it can be approximately computed from the expression log(A/AQ) = 1.48 — 2.28(B — V). (5.25)

Thus given B-V and age, the rotation period can be calculated from eqn (5.24).

In Table 5.6 are given the rotation periods and estimated ages, for stars with ages ranging from 0.1 to 8.5 Byr, from the Sun in Time programme using different age-indicator techniques. The rotation period based on eqn (5.24) was fitted to the observations with a value of m = 0.52 ±0.02, with the standard error at the 95% confidence level and normalized to the present solar value. The computed rotation period is compared with that based on observations in Fig. 5.6. Also shown is the power-law fit P = PQ(t/tQ)n, where n = 0.60 ±0.04, in agreement with the rotation period-age relation.

5.5.3 Rossby number with spectral type and age

The convective overturn time, tc, for G, K and M stars can be computed from the expression of Noyes et al. (1984)

log(Tc) = 1.362 — 0.166x + 0.025x2 — 5.323x3 x> 0, (5.26)

where x =1 — (B — V). Thus, given the stellar rotation period and B-V the Rossby number can be computed from eqn (5.23). The stellar B-V (available from the SIMBAD database), effective temperature, radius, convective overturn time and Rossby number are given in Table 5.7. The variation of the Rossby

Stellar Age (Gyr)

FlG. 5.6. Stellar rotation period versus age for solar-like stars. Empty circles are based on observations while full squares are computed values using eqn (5.24). The solid line is the power-law curve P = [email protected](t/[email protected])0-6. (Vardavas 2005)

number with stellar age is shown in Fig. 5.7. We note that the Rossby number of 18 Sco does not follow the trend of the others and its rotation period seems low for its estimated age. One can also compute the Rossby number from the stellar age and B-V from eqn (5.24) according to as shown in Fig. 5.7. This is useful for studies of the evolution of planetary atmospheres given the evolution of stellar B-V with age. Furthermore, a simple power law fits the present data of form

5.5.4 XUV-Lya emission and Rossby number

A parameter that is commonly used to give a measure of stellar chromospheric activity is denoted by RHK. It is defined as the ratio of the emission flux, FHK,

FlG. 5.7. Stellar Rossby number computed using eqn (5.23) (empty circles) versus stellar age. The solid line is the power-law fit Kq (t/tg)0'5. Solid squares are values calculated using eqn (5.28). (Vardavas 2005)

Stellar Age (Gyr)

Fig. 5.8. XUV(1-1200 A) stellar surface emission flux ratios versus age for solar-like stars. Empty circles are based on measurements, while full squares are computed values using the stellar Rossby number and effective temperature in eqn (5.33). The solid line is a (t/tg)-1'25 power-law fit. (Vardavas 2005)

Star |
B-V |
TeS |
R/RQ |
Tc |
K |

EK Dra |
0.59 |
5870 |
0.94 |
8.54 |
0.31 |

n1 UMa |
O.BS |
BSB0 |
0.97 |
7.99 |
0.61 |

X1 Ori |
0.B9 |
BS90 |
1.00 |
S.B4 |
0.61 |

9 Cet |
0.67 |
B740 |
1.00 |
13.14 |
0.BS |

K1 Cet |
0.6S |
B7B0 |
0.93 |
13.71 |
0.67 |

ß Com |
0.B7 |
6000 |
1.0S |
7.4B |
1.61 |

lB Sge |
0.61 |
BSB0 |
1.09 |
9.67 |
1.40 |

Sun |
0.6B |
B777 |
1.00 |
11.99 |
2.1S |

lS Sco |
0.6B |
B7SB |
1.02 |
11.99 |
1.92 |

ß Hyi |
0.62 |
B774 |
l.S9 |
10.24 |
2.73 |

16 Cyg Â |
0.64 |
B790 |
1.27 |
11.40 |
3.07 |

Star |
XUV (1 AU) |
XUV (surface) |
Lya (1 AU) |
Lya (surfai |

EK Dra |
110.7 |
135.8 |
14.6 |
17.9 |

n1 UMa |
27.9 |
33.9 |
6.S |
S.3 |

X1 Ori |
27.9 |
33.9 |
6.S |
S.3 |

9 Cet |
10.3 |
12.4 |
4.4 |
B.3 |

K1 Cet |
11.0 |
13.2 |
4.7 |
B.6 |

ß Com |
3.B |
4.0 |
2.2 |
2.B |

1B Sge |
2.9 |
3.3 |
1.9 |
2.2 |

Sun |
1 |
1 |
1 |
1 |

1S Sco |
0.90 |
0.S7 |
0.96 |
0.92 |

ß Hyi |
0.63 |
0.B2 |
0.76 |
0.63 |

16 Cyg Â |
0.B0 |
0.34 |
0.63 |
0.43 |

from the cores of the Ca II H and K lines to the total bolometric emission (ffTff) of the star

and related to the Rossby number by an expression of the form

We shall choose the following simpler power-law dependence on Rossby number for the XUV (1-1200 A) and Lya activity, defined as the ratio of the total emission flux within these spectral regions to the total bolometric emission of the star

100 n

Stellar Age (Gyr)

FlG. 5.9. Lya stellar surface emission flux ratios versus age for solar-like stars. Empty circles are based on measurements while full squares are computed values using the stellar Rossby number and effective temperature in eqn (5.33). The solid line is a (t/tg)-0'78 power-law fit. (Vardavas 2005)

The ratio of the stellar surface (in fact chromospheric layers and coronal layers close to the star) emission flux (Fs in erg cm~2 s_1) relative to the corresponding present solar mean value (Fsq), defined by E = Fs/Fsq, can then be calculated for the XUV and Lya from

From observations by the ASCA, ROSAT EUVE, FUSE and IUE satellites, Lammer et al. (2004) derived, for the stars of Table 5.6, stellar/solar XUV (1-1000A) emission flux ratios, while Ribas et al. (2005) derived improved XUV (1-1200A) flux ratios for six stars (EK Dra, n1 UMa, x1 Ori, k1 Cet, 3 Com and 3 Hyi) at 1 AU and scaled to the expected solar radius at the star's age based on the evolution of a 1 solar mass star with solar metallicity. Similarly, from high-resolution spectroscopic observations by the Hubble Space Telescope, Lya emission flux ratios were derived for the stars of Table 5.6. The measured XUV and Lya flux ratios at 1 AU are given in Table 5.8, taking into account the improved values for the stars mentioned above. These scaled ratios (F/Fq) at 1 AU are related to the corresponding ratios based on fluxes at the stellar and solar surface via

where R©(t) is the expected solar radius at the star's age based on the evolution of a 1 solar mass star with solar metallicity. In Figures 5.8 and 5.9 the emission flux ratios at the stellar surface are compared with those derived from the stellar Rossby number and effective temperature, using eqn (5.33) and data from Table 5.7, with 3 = 2.46 ±0.12 for the XUV and 3 =1.52 ±0.07 for the Lya.

If we insert the power-law expression, eqn (5.29), for K in eqn (5.33) with the appropriate values for 3 we obtain a one-parameter fit to the flux ratio values at the stellar surface

1.25

Teff

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