Situación nutricional

4.5.1

Recent Observations

Table 4.4: Top: Stellar parameters of objects recently observed with CARMA, and APEX. The parameters are collected from various sources, as annotated. Bottom:

The observed flux densities.

Name HD No. Teff (K) φ (mas) Parallax (mas) Fe/H log(g) F

α Taua _{29139 3733 21.1 48.94 -0.1 1.2 1.96} α Booa _{124897 4630 21.37 88.83 -0.32 1.5 2.7} β Anda _{6860 3800 13.75 16.52 -0.04 1.5 2.45} µGem a _{44478 3450 15.12 14.08 -0.09 1.00 1.75} α Ceta _{18884 3675 13.24 13.09 -0.5 0.5 1.73} β Gru 214952 3500b _27.80b _18.43c _0.0d _0.4d _1.0d γ Cru 108903 3626b _26.37b _36.83c _0.0e _2.0e _1.4f g Her 148783 3250g _18.4h _9.21c _-0.01g _0.20g _1.0h a_{− Parameters as in Table 4.3} b_{− Engelkeet al. (2006)} c_{− van Leeuwen (2007)}

d_{− Judge (1986)} e_{− Carpenter & Wahlgren (1990)} f_{− P´erez Mart´ınez et al.}

(2011) g_{− Soubiran et al. (2010)} h_{−Luttermoser et al. (1994b)}

Name CARMA CARMA APEX

1 mm Flux (mJy) 3 mm Flux (mJy) 345 GHz Flux (mJy)

α Tau 47.6 _± 2.2 13.89 _± 0.32 112.91 _± 12.36 α Boo 51.1 ± 1.8 21.48 ± 0.75 138.97 ± 23.37 β And 14.8 ± 1.2 6.94 ± 0.29 ... µGem ... 5.66 _± 0.36 ... α Cet 15.3 ± 1.1 5.15 ± 0.19 ... β Gru ... ... 163.59 ± 9.08 γ Cru ... ... 132.68 _± 15.87 g Her ... 5.66 ± 0.36 ...

Recently observations were made using CARMA and APEX of 8 K and M giants. The CARMA observations were made at two frequencies, 100 GHz and 225 GHz, and the APEX observations were made at 345 GHz. These data were reduced and provided by O’Gorman (2014). The data and the stellar parameters of the targets are tabulated in Table 4.4. The errors included in this table are

3000 3500 4000 4500 5000 T_eff (K) -4000 -2000 0 2000 4000

Flux Scatter [Obs - Model] (K)

α Tau α Tau α Tau α Boo α Boo α Boo β And β And µ Gem α Cet α Cet β Gru γ Cru g Her CARMA 1mm CARMA 3mm APEX

Figure 4.7: M2 model compared with high S/N CARMA and APEX observations. We see that the model again provides a good fit, with most objects falling within

±30% at all frequencies.

1-σerrors and do not include the 10% systematic uncertainty, however all analysis presented hereafter is based on the data with 1-σ errors and the 10% systematic uncertainty due to calibration summed. These data were compared with the M2 model (which we recall had the lowest χ2_{), and the results of that calculation} can be seen in Fig. 4.7. g Her is a clear outlier in this sample, and this may be accounted for by its spectral type (M6 III), for much the same reasons as R Lyr in the previous sample, noting that Luttermoseret al. (1994a) could not construct a time-independent, semi-empirical model for this object.

We see that the model is relatively consistent and reproduces these brightness temperatures quite well, though with the same inherent scatter. It is important to bear in mind that the model may be reaching the limits of its validity for the higher frequency APEX observations (central wavelength = 869µm (Lapkin et al., 2008)). This is another constraint on the model, as we enter the sub-mm/µm

Table 4.5: Spectral indices from the CARMA observations, and computed from M2. Name α1mm−3mm αmodel α Tau 1.52 ± 0.33 1.37 α Boo 1.07 _± 0.34 1.60 β And 0.93 _± 0.31 1.47 µ Gem ... 1.27 α Cet 1.34 _± 0.39 1.28 β Gru ... 1.18 γ Cru ... 1.32 g Her ... 1.22

wavelength regime our assumptions may not remain valid. This may be due to bound-free emission/opacity, which are not accounted for in our model, becoming important considerations in the sub-mm regime (Chluba & Sunyaev, 2006). There may be a contribution from dust emission, however the majority of these objects are thought be relatively dust-free (Sutmann & Cuntz (1995),O’Gorman et al. (2013)), and any dust emission would be expected to peak at somewhat shorter wavelengths (peaking closer to 100µm (Draine, 2011)).

4.5.2

Spectral Indices

The spectral index is the power law exponent which relates the flux density and the frequency; Fν ∝να. For an isothermal, non-extended, optically thick atmosphere α will follow the Rayleigh-Jeans tail of the Planck function, α = 2, and in the case of an optically thin plasmaαwill have the same frequency dependence as the Gaunt factor,α =−0.17. In a plasma with temperature and density gradients, α

will be intermediate between these two values. In reality it is possible to have a value forα which lies outside of this range as a result of the fact that stellar radius varies as a function of frequency. It is due to precisely the effect of seeing a larger, hotter stellar surface at lower frequencies that results in α not simply equalling 2 for the optically thick emission which comprises much of our sample (Kundu, 1965; Newell & Hjellming, 1982).

Figure 4.8: Spectral indices, computed at 225GHz and 100GHz, for a range of

Teff and log(g). We see that at high temperature and low gravity the spectral index index matches the optically thin limit, and as log(g) increases the optically thick case is recovered.

Table 4.5, where:

α= log10(F225/F100) log10(225×109/100×109)

(4.45) The computed values appear to match quite well with observation, indicating that the material probed at different frequencies may have the same temperature variation as our model.

Analytically, we can determine the expected spectral index for given power- law varying temperature and electron density, ne ∝r−p and T ∝r−n (Seaquist &

Taylor (1987),O’Gorman et al. (2013)):

α= 6.2−4p+ 3.45n

1₋2p+ 1.35n (4.46)

our case, wherene is assumed to be constant,p= 0, andT is assumed to be linear

in height, n =−1. This gives α= 1.17, which is generally in agreement with the values measured and computed.

In Fig. 4.8 the values of α returned for a range of Teff and log(g) are plotted. We can see that our model predicts that for low gravity objects of intermediate temperatureαwill match the optically thin limit, and as log(g) increasesαreflects the optically thick case. It is interesting to note that this seems to be most strongly a function of log(g), allowing us to differentiate an object of high gravity from one of low gravity solely by reference to their mm spectral index.