University of Amsterdam
MSc Physics & Astronomy
Track Astronomy & Astrophysics
Master’s Thesis
Constraining starspot fraction and temperature in
weak-lined T Tauri stars using spectroscopy
by
J. H. Bootsma, BSc 10251499
October 2018
60 ECTS
Research conducted between September 2017 and October 2018
Supervisor:
Prof. dr. Carsten Dominik Daily supervisor:
Dr. Melissa K. McClure
Examiner: Dr. Jayne L. Birkby
A B S T R A C T
Context. Whether the gas situated in the innermost regions of a protoplanetary disk (PPD) is optically thin or optically thick determines the radial distance upward of which rocky planets (or planetary cores) can form. Obtaining information about the opacity of the inner gas is, however, impeded by a number of phenomena. One of which is the presence of starspot on the surface of stars that host PPDs. These starspots have similar temperatures as the inner gas and this makes the source of (some of) the light we receive from such star-disk system ambiguous.
Aims. We aim to constrain the starspot fraction ( fspot) and the starspot temperature (Tspot)
of two weak-lined T Tauri stars (WTTS). Also, we aim to deliver a set of absorption lines that can be applied to perform this analysis on classical T Tauri stars, the hosts of PPDs.
Methods. We select three absorption lines that (1) heavily depend on both the temperature (T) and the surface gravity (log g) around the expected starspot temperature and (2) mildly depend on T and log(g) around the effective temperatures of the WTTS in our sample. We construct two-temperature model atmospheres by combining two single-temperature PHOENIX models using a certain fraction such that each model spectrum has a Te f f, log(g),
Tspotand fspot. We fit these model spectra to the selected line profiles.
Results. We select the K I lines centered at 1.169, 1.17751 and 1.2525µm as starspot-sensitive and we select PZ99 J160550.5-253313 and RX J0457.5+2014 as the spottiest targets in our sample. The best-fitting parameters [Tspot, fspot] for our two targets are [3700K, 100%] and
[2967K, 83%]. The results are, for each target, consistent over the three different spectral lines. The spot temperatures indeed overlap with the temperature of the inner gas. The 83% spot coverage can be explained by an inclined stellar axis. The 100% spot coverage, how-ever, is most probably due to a methodological shortcoming. Before applying this method to CTTS, radiative transfer modeling is required to exclude that the selected absorption features will be filled in by emission from the hot inner disk.
C O N T E N T S
1 i n t r o d u c t i o n 1
1.1 Low-mass star formation and evolution 1 1.2 PPDs & planet formation 4
1.3 Observing TTS and their disks 7 1.3.1 Spectral energy distribution 7 1.3.2 Emission of star-disk systems 7 1.3.3 Starspots 8
1.4 This work 8 2 m e t h o d s 10
2.1 Selection of spectral lines 10
2.1.1 The depth of an absorption line: equivalent width 10 2.1.2 Temperature dependence of the equivalent width 11 2.1.3 Surface gravity dependence of the equivalent width 13 2.2 Identification of sources 15
2.3 Construction of W
λ vs. SpT plots 17
2.4 Shaping the models & the data 18 2.4.1 Shaping the models 18 2.4.2 Shaping the data 21 2.5 Fitting models to data 21 3 r e s u lt s 23
3.1 Line and target selection 23 3.2 Fit results 25
3.3 Fitting PZ1625 data 25
3.3.1 Fitting PZ1625 data with models with log(g) =3.5 25 3.3.2 Fitting PZ1625 data with models with log(g) =4.0 30 3.4 Fitting RX0420 data 34
4 d i s c u s s i o n 40
4.1 The constrained spot fractions and temperatures 40 4.1.1 Discussion of the found starspot fractions 40 4.1.2 Discussion of the found starspot temperatures 41 4.2 Reviewing the method 43
Contents iv
4.2.1 Consistency of the results over the different lines 43 4.2.2 Temperature dependence of W
λ 44
4.2.3 Importance of the ‘correct’ spectral type and Te f f 44 4.2.4 Coarseness PHOENIX model grid 45
4.3 Applying this method to CTTS 46 5 s u m m a r y & conclusions 49 a s a m p l e o f l i n e s 59
b e q. width versus spectral type plots spex data 61
c e q. width versus spectral type plots x-shooter data 67 References 73
1
I N T R O D U C T I O N
1.1 l o w-mass star formation and evolution
The process of low-mass star formation (< 2M ) is well explained by the widely accepted
theory of the collapse of a rotating, giant molecular cloud (GMC). A GMC is caused to gravitationally collapse, because it exceeds the Jeans mass (MJ), or because a local instability
occurs. The cloud will fragment into multiple denser regions (Class 0 in Fig. 2). The center of such a denser region becomes optically opaque, temperature continues to increase due to further contraction of the cloud and due to shocks created by material that is falling onto the central region. Convection and radiation from its exterior allow for further contraction up to the moment where the internal pressure can support against the gravitational pressure (Larson 1969). At this point, a protostar has formed. A protostar is surrounded by a circumstellar disk and an envelope of gas and dust through which it accretes material (Class I in Fig. 2). At the end of this stage the envelope is dispersed and the stellar object has nearly acquired all of its mass. At this point, the stellar object is called a pre-main-sequence star (PMS) which is still surrounded by a circumstellar disk (Class II in Fig. 2). In our case (M? <2M ) these are called T Tauri stars (TTS).
The circumstellar disk is often called a protoplanetary disk (PPD) since this is believed to be the ancestor of a planetary system and the birthplace of planets. Within a couple of million years the PPD depletes its mass by accreting gas onto the star (Armitage 2011) and by the photoevaporation of gas (Gorti et al. 2015). In the same time dust grains can grow up to planetesimals, Earth-like planets and the rocky cores of giant gas planets (Lissauer et al. 2009). What remains is the TTS surrounded by a gas-poor debris disk (Class III in Fig. 2).
TTS are young F-, G-, K- or M-type stars with masses varying from 60MJup to 2M .
TTS are identified by optical variability and strong chromospheric spectral lines. They are divided into two sub-groups: classical T Tauri stars (CTTS) and weak-lined T Tauri stars (WTTS). Whether a TTS is defined as WTTS or CTTS is determined by their amount of
1.1 low-mass star formation and evolution 2
Hα emission. CTTS have more Hα emission than WTTS (see e.g. White & Ghez (2001) or White & Basri (2002) for a detailed description of how to distinguish CTTS from WTTS). An important difference between CTTS and WTTS is that CTTS show signatures of accretion whereas WTTS lack these (see Fig. 1). This suggests that a circumstellar disk is still present around CTTS (roughly corresponding to Class II in Fig. 2), but that the (inner) disk around WTTS has depleted (roughly corresponding to Class III in Fig 2).
Figure 1.: Schematic representation of a CTTS (left) and a WTTS (right). The arrows in the left diagram repre-sent the accretion. The emission induced by the accretion shock is also indicated. Figure from Herbst (2012).
1.1 low-mass star formation and evolution 3
Figure 2.: Diagram showing the different stages of the formation of a planetary system. The left column shows the corresponding spectral energy distribution we would observe. CTTS roughly correspond to Class II and WTTS roughly correspond to class III. Figure from Andre (2002).
1.2 ppds & planet formation 4
1.2 p p d s & planet formation
A protoplanetary disk is a composition of gas and dust (rocky material) rotating around its host star (see Fig. 3 for an overview). The general consensus is that planet formation in a protoplanetary disk is governed by the following sequence of events: coagulation of sub-micrometer dust particles (which are present throughout the universe in the interstellar medium) (e.g. Dubrulle et al. (1995), Dullemond & Dominik (2005); streaming instabil-ity (creates clumps that become gravitationally unstable, e.g. Youdin & Goodman (2005), Sch¨afer et al. (2017); pebble accretion (e.g. Ormel & Klahr (2010), Visser & Ormel (2016); followed by the core-accretion model of Pollack et al. (1996).
However, observations of various types of planets at various distances from the star requires various specific formation procedures. In our own solar system we have small, terrestrial planets up to roughly 1 AU and gas giants further out. In other planetary systems we have been able to detect massive close-in planets such as Hot Jupiters, super-Earths and gas giants. Fig. 4 shows the radial distribution of confirmed exoplanets with M>10MEarth.
It is important to note that our exoplanet detection techniques, the wiggling of a star due to the gravity of a planet and starlight being temporarily blocked by a passing through planet, are biased towards close-in and more massive planets (e.g. Marcy et al. (1998); Charbonneau & Deming (2007). However, the fact that these planets are situated so close to their host star is indisputable and raises the question whether they formed further out and drifted inwards or they formed in situ (e.g. Boley et al. (2015); Dawson & Johnson (2018).
Figure 3.: Cross-section of a PPD (Dullemond & Monnier 2010). Indicated is the dust inner rim: the natural boundary up to which rocky planets and planetary cores can form.
1.2 ppds & planet formation 5
Figure 4.: Histogram of the orbit semi-major axis of confirmed exoplanets with M > 10 MEarth.
Note the amount of planets within 0.04AU. Figure from the NASA Exoplanet Archive (https://exoplanetarchive.ipac.caltech.edu/).
How close to the star planets can form, is determined by the radius of the ‘dust inner rim’. Inside this virtual boundary, dust sublimates. The location of the dust inner rim is thus defined by the local temperature of the disk: the radial location where T =Tsubl. The
sublimation temperature is about 1000-1800K (eg. Natta et al. (2001), Dullemond & Monnier (2010) depending on the local pressure (the sublimation temperature increases with pres-sure). The radial temperature distribution in the (inner) disk depends on the luminosity of the star. For more luminous stars, temperatures will be higher at a certain radial distance, compared to less luminous stars. Anthonioz et al. (2014) have used VLTI/PIONIER NIR interferometry to resolve a number of PPDs around TTS and modeled the visibility data using two different models. They find sublimation radii of∼0.04-0.2AU. Again, this is the natural boundary up to which rocky planets can form.
However, optically thick gas might be present in the innermost region (Monnier & Millan-Gabet 2002; Eisner et al. 2004; Monnier et al. 2005; Kraus et al. 2009; Benisty et al. 2010; Dullemond & Monnier 2010; Fischer et al. 2011) and this would shield the region behind it from stellar radiation (this region is indicated by steel blue in the center of Fig. 5). This idea is supported by Fig. 6 (from Anthonioz et al. (2014) that shows that a number of disks have inner rim radii smaller than expected from the luminosity of its host star.
The consequence of this idea is that dust will remain solid at shorter distances from the star than thought before, and that in situ formation is a possible explanation also for exoplanets situated closer in than the 0.04AU found by Anthonioz et al. (2014). Looking at Fig. 4 this includes 75−150 exoplanets. Also, Boley et al. (2014) found that formation of planets at the dust sublimation radius is favoured due to enhanced stickiness of partly molten (i.e. not solid anymore, but not yet sublimated) dust. This stresses the importance of knowing the location of the dust sublimation radius even more.
1.2 ppds & planet formation 6
Figure 5.: Cross-section of inner few AU of PPD (adjusted figure from Dullemond & Monnier (2010). It shows how optically thick gas would shield the region behind it from stellar radiation, cooling this region below the dust sublimation temperature. Also, note how the temperature of the starspots (2000-4500K) overlaps with the temperature of the inner gas (1800-6000K, Muzerolle et al. (2004).
Figure 6.: Plot showing the inner rim locations as a function of stellar luminosity. The blue lines represent the sublimation radius as a function of the stellar luminosity for grains sublimating at 1500 K (straight line) and 2000 K (dashed line). The blue triangles (including error bars) show inner rim locations calculated with a model consisting of thermal emission and scattered light by Anthonioz et al. (2014). Interesting to see is that, based on temperature, the inner rim location too close to the star for a number of star-disk systems.
1.3 observing tts and their disks 7
1.3 o b s e r v i n g t t s a n d t h e i r d i s k s
1.3.1 Spectral energy distribution
Collecting light from an object in the universe is what astronomers call observing. From this light we aim to deduce properties of its source as well as properties of the medium that the light has interacted with on its way to the telescope. A diagram that shows the amount of light that is emitted per wavelength is a spectral energy distribution (SED). The y-axis shows the energy of the light (λFλ) and the x-axis the wavelength (λ). The left part of Fig. 2 shows simplified examples of such a diagram, together with the corresponding star forming stages. SEDs give a lot of information about the processes that are taking place in the system, because each phenomenon that produces light in the plotted wavelength range is represented in the SED. However, separating different sources of light is complicated once multiple processes produce light with the same wavelength. When thinking of an SED, think about the light being delivered as if in a big bucket that only separates based on wavelength.
1.3.2 Emission of star-disk systems
A star-disk system contains a variety of temperature regions (see Fig. 5). Each temperature region is essentially an individual source of light. The light emitted per area by such a region (in W·sr−1·m−3) is roughly that of a black-body, described by the Planck function
(see equation 1.1, where h is Planck’s constant, c is the speed of light and kBis Boltzmann’s
constant). Examples of such light sources are: the non-spotted surface of the star, the spotted surface of the star, the (optically thick) gas present in the inner disk and the dust inner rim. The parameters of interest of a certain temperature region are (1) the value of the temperature and (2) the emitting area, as these together determine the amount of light emitted. Fig. 7 shows how we can reconstruct data by fitting it with different areas having different temperatures (the dotted lines).
Bλ(λ, T) = 2hc2 λ5 1 eλkB Thc −1 (1.1) This thesis focuses on determining the starspot temperature and the fraction of the stellar surface that is covered with starspots.
1.3 observing tts and their disks 8
Figure 7.: Example of how the light of different sources of light can combine into an SED (from Fischer et al. (2011). It shows the emission excess above the stellar SED of BP Tau between 0.8 and 2.1 µm. The dotted lines show the contribution of three temperature components and their filling factor (where
f =1 corresponds to the surface of the star). Temperature values are: 8000K, 2500K and 1400K with
filling factors 0.03, 1.5 and 16, respectively. The dashed line shows the sum of the three components. The filling factors exceed 1.0 (corresponding to the full surface of the star), because the surface of the disk around the star is also considered.
1.3.3 Starspots
Starspots are cooler patches on the surface of a star as a result of strong magnetic fields blocking the outflow of energy through the stellar surface (Biermann 1938). The tempera-ture difference between the starspots and the stellar surface varies with the spectral type of the star (i.e. its surface temperature, Te f f): Te f f −Tspot is about 200K for late M-type stars
(Te f f = ∼2800K) and Te f f −Tspotcan go up to 2000K for early G-type stars (Te f f = ∼6000K)
(e.g. Vrba et al. (1988); Berdyugina (2005); Grankin et al. (2008); Gully-Santiago et al. (2017). The first report of starspots date from 1611 by David and Johannes Fabricius (Mitchell 1916). They discovered that the sun is subject to starspots. While the sun is the closest star that carries starspots, other stars are much heavier spotted (e.g. Hall (1972); Gully-Santiago et al. (2017). As stated before, young stars (like TTS) are convective. Starspots are a visible component of magnetic flux tubes through the convective zone of a star (see Fig. 8).
Currently, the most advanced method to study starspots is referred to as Doppler imag-ing. This method, however, requires time-series of well-sampled high-resolution spectra, which are difficult to obtain (K ¨unstler et al. 2015).
1.4 this work 9
Figure 8.: Left: NASA picture of the sun showing sunspots. Right: Explanatory diagram of the connection between convection, magnetic field lines and starspots (applied to the sun). Figure created by Artem Cheprasov, obtained from https://study.com/academy/lesson/why-do-sunspots-occur.html.
1.4 t h i s w o r k
In this thesis we select two WTTS of which we constrain the fraction of the surface that is covered with starspots and the corresponding starspot temperature.
In order to do this, we select spectral lines that are (ideally only, in practice partly) pro-duced in the cooler areas of the photosphere. For the selection of the spectral lines and for the selection of the targets we compare the depth of a set of absorption lines for a sample of WTTS with that of dwarf and giant standard stars of equal spectral type (i.e. surface temperature). For the selected lines we fit the line profile of the two spottiest WTTS in our sample with synthetic spectra that are derived from two-temperature model atmospheres. We create these models by combining two, single-temperature PHOENIX models. Using a least-square method we investigate which combination of starspot fraction and starspot temperature best represents the data.
For this research we use WTTS, because this method has not yet been fully developed. By using WTTS we exclude the possibility that the inner disk that surrounds a CTTS and has similar temperatures as the starspots has a contribution to the line profile. We aim to select a set of lines that eventually can be applied to CTTS data to determine also for these sources what fraction of the surface is covered with starspots and what the starspot temperature is.
2
M E T H O D S
The Methods section of this thesis is divided into four parts. Firstly, we describe the method-ology of selecting which spectral lines are suitable for determining the starspot fraction and temperature of a TTS. Secondly, we explain how we identify the WTTS sources we will fo-cus on. Thirdly, we walk through the required steps of how we shape the PHOENIX models and the data in order to do our analysis. Finally, we describe how we compare the models with the data and how we, in the end, will draw our conclusions.
2.1 s e l e c t i o n o f s p e c t r a l l i n e s
For this work we use high-resolution spectroscopic measurements in the near-infrared. We have spectra of 17 WTTS that were taken by Manara et al. (2017) using the X-shooter spec-trograph on the Very Large Telescope. An example of such a spectrum is shown in Fig. 9, including a zoom in on a spectral line. These spectral lines are formed as a result of absorption of continuum emission in the photosphere (for an explanatory diagram see left panel of Fig. 10). We use these absorption features to constrain the spot temperature and the spot fraction of a WTTS. However, not every absorption line is suitable for this. To de-termine which of the absorption lines in our sample of 54 lines serve our purpose, we use the dependency of the depth of the spectral line (Wλ) on temperature (T) and on surface
gravity (g). Before getting to these dependencies, however, it is worth discussing how we quantify the depth of an absorption line.
2.1.1 The depth of an absorption line: equivalent width
To quantify the depth of an absorption line centered at wavelength λ we use the equivalent width, Wλ[ ˚A], introduced by Minnaert et al. (1940). It is a measure for the line profile
integrated depth and is equivalent to the width of a rectangular line profile that goes down to a flux level of zero (maximum absorption) and has the same surface area as the original
2.1 selection of spectral lines 11
line profile. Its definition is shown in Eq. 2.1, whereFλ and FC are the line flux and the
continuum flux, respectively. The right panel of Fig. 10 shows a more intuitive, visual representation of the equivalent width.
Wλ = Z λ 1− Fλ FC dλ (2.1)
Figure 9.: NIR spectrum of target PZ1625 taken with X-shooter. The vertical lines indicate the wavelengths of the spectral lines we ultimately identify as starspot-sensitive (see Section 3).
2.1.2 Temperature dependence of the equivalent width
Since WTTS have photospheres that have different temperature components (a relatively hot Te f f and a relatively cold Tspot), it is key to understand how the equivalent width of different
absorption lines depends on the temperature. Absorption lines form, because photons (with a specific wavelength) coming from a bright background are absorbed. Which photons are absorbed (i.e. which wavelength) depends on the energy state of the absorbing atoms and molecules. The energy state of a gas is described by Boltzmann’s equation and Saha’s equation (Eqs. 2.2 and 2.3, respectively), which both depend on the temperature of the gas. Boltzmann’s equation describes the excitation state of the gas and Saha’s equation describes the ionization state of the gas. In these equations N and n correspond to the number of particles in a certain state, g is a factor that corresponds to the degeneracy of a certain state,
2.1 selection of spectral lines 12
Figure 10.: Left: a schematic representation of how absorption lines are produced (this is an example of the Sun). In the photosphere the temperature decreases in the outward direction and an observer therefore sees a less bright layer in front of a bright layer (the background). This causes the formation of absorption lines in the spectrum. Figure from http://www.universe-review.ca, section Stars/Stellar structure. Right: A diagram visualizing the relation between the depth and the equivalent width of an absorption line. The surface area of the shaded rectangular is equivalent to the line profile integrated depth. Figure made by Dorottya Szam, taken from Guzman (2017).
E and e are the energies of a certain state, kB is Boltzmann’s constant and T is the local
temperature. Nhigh Nlow = g2 g1 e− Ehigh−Elow kBT (Boltzmann) (2.2) ni+1ne ni = 2 λ3 gi+1 gi eei +1−ei kBT (Saha) (2.3)
In a stellar photosphere a number of different atoms are present and each atom has its own many different possible energy transitions, dependent on the temperature. This makes the temperature dependence of Wλ rather complex. Therefore we empirically investigate
how the equivalent width of each individual absorption line depends on temperature: we measure the equivalent width of a particular line in spectra of stars with different effective temperatures (i.e. photospheric temperature) and show the results in a plot. This gives us a visual representation of how the equivalent width of that particular line changes with temperature. As laboratories, we use dwarf stars (spectra from the IRTF Spectral Library of dwarf and giant standard stars (Rayner et al. 2009) with effective temperatures varying from 2700up to 7000K. Fig. 11 shows what spectral type (i.e. effective temperature) dependence we are looking for: a weak temperature dependence around the effective temperature of the star and a strong temperature dependence around the expected starspot temperature (recall that this is 500−2000K cooler than the effective temperature). A spectral line that
2.1 selection of spectral lines 13
has an equivalent width that depends on temperature in this manner (and thus similar to the dependence shown in Fig. 11), is affected by the cooler areas of the surface rather than the hotter areas and therefore ‘traces’ these cooler starspots.
Figure 11.: An example of what the equivalent width vs. spectral type plot would look like for a line that is sensitive to starspots.
2.1.3 Surface gravity dependence of the equivalent width
Besides temperature, we also use the dependence of the equivalent width on surface gravity (g). The surface gravity of a star is defined by its mass and its radius (see Eq. 2.4, G is the gravitational constant). g= GM R2 = G R2
∑
i nimi (2.4)When a star has a higher surface gravity, the pressure at the surface of the star is higher. A higher pressure results in broader emission lines through a mechanism that is in gen-eral called ‘pressure broadening’. In the case of late-type, cooler stars (Solar and cooler), this type of broadening is dominated by what is called ‘Van der Waals’ broadening: the perturbation of energy levels due to interactions of non-hydrogen like atoms with neutral hydrogen. This mechanism is dependent on the local number density, because having more
2.1 selection of spectral lines 14
particles in the same volume results in a higher probability of collisions and close encoun-ters and thus a higher probability of perturbed energy levels. Since a higher surface gravity means that the same mass must be present within a smaller radius, the number density is in this case also higher (Eq. 2.4). Wrapping up these dependencies, the broadening of emis-sion lines increases with the surface gravity of the star. Therefore, a higher surface gravity results in a higher equivalent width. An example of this relation is shown in Fig. 12 where the profile of an absorption line is plotted for multiple values of log(g). It is evident that the area associated with the absorption line increases with surface gravity.
Figure 12.: Figure from Gray (2005) showing the deformation of an absorption line under the influence of a changing surface gravity.
In this work, we measure the equivalent width of our sample of spectral lines not only in the spectra of dwarf stars, but also in the spectra of giant stars (also from the IRTF Spectral Library of dwarf and giant standard stars (Rayner et al. 2009). Dwarfs and giants are in a different evolutionary stage: dwarfs are main-sequence stars, whereas giants have expanded drastically since their dwarf phase. Due to this difference in radius dwarfs have a much higher surface gravity than giants (see Eq. 2.4): log(gdwar f) = ∼4.5 while log(ggiant) =
∼1.5 (Carroll & Ostlie 2017). So for spectral lines that are sensitive to surface gravity in the way discussed above, this means that the equivalent width of a line measured in spectra of dwarf stars is, for every temperature, at least as high as the equivalent width of that same line measured in spectra of giants. The third type of stars we use (the ones we are interested in), WTTS, are pre-main-sequence stars. This means that they are contracting towards the dwarf evolutionary stage. Therefore TTS have an intermediate surface gravity (log(g) = ∼3.5−4.5), compared to dwarfs and giants. In other words: there is a defined
2.2 identification of sources 15
region in the equivalent width vs. spectral type plots where one would expect the WTTS to end up. Fig. 13 shows an example of such a plot; the region where one would expect WTTS to end up is indicated with grey shading.
Figure 13.: Same as Fig. 11, but now distinguished between the dwarf and the giant trend. The shaded region
is, based on the log(g)dependence, where one would expect a WTTS with a certain temperature to
show up (two examples are shown).
2.2 i d e n t i f i c at i o n o f s o u r c e s
To understand which sources from our sample we want to investigate, there is one more effect that needs addressing: the effect that starspots have on the placing of a WTTS in an equivalent width vs. spectral type plot. We will ultimately use this effect to select two WTTS for which we will constrain the spot fraction and the corresponding spot temperature.
The horizontal placement of a WTTS from our sample is defined by the spectral type assigned by Manara et al. (2017). This spectral type represents an effective temperature. As discussed before, starspots have temperatures that are 200−2000K lower than the effective temperature. This means that it, in principle, is possible to have a WTTS that has a tem-perature distribution such that the hot regions have a temtem-perature at which the equivalent width of a spectral line is not very sensitive and the colder regions have a temperature at which the equivalent width of a spectral line is very sensitive. If one would measure the equivalent width of a starspot-sensitive line in this WTTS and this WTTS has an
appropri-2.2 identification of sources 16
ate spectral type, it would appear above the dwarf trend (see the left panel of Fig. 14 for an explanatory figure and the right panel for a plot produced using actual data (McClure et al. 2013). The higher a WTTS appears above the dwarf trend, the bigger the effect of the starspots is on the equivalent width. In other words: in order to select the spottiest targets, we want to select the WTTS that lie the furthest above the trend that the dwarf stars follow in the equivalent width vs. spectral type plots of the lines we will select.
Figure 14.: Left: Same as Fig. 13, but additionally showing where spotted WTTS with spectral types earlier than M0 show up on these kind of plots (for starspot-sensitive lines). Note how the WTTS with spectral type K0 appears above the dwarf trend. This, because absorption happens in the starspots on its surface. Right: Figure from McClure et al. (2013) showing the effect that starspots can have on where TTS sources show up in an equivalent width vs. spectral type plot. The TTS sources are indicated by the red crosses, the dwarfs are indicated by the black dots (including a trend line, black solid line) and the giants are indicated by the grey squares (also including a trend line, grey solid line).
In summary
Summing up the selection criteria: we need spectral lines that have a strong Wλ vs. SpT
dependence at temperatures around Tspotand, in addition, we want the equivalent width of
these lines to be positively correlated with log(g), such that there is a defined region where we expect the WTTS to show up in the Wλvs. SpT plots. Furthermore, we select the targets
2.3 construction of W
λ v s. S p T plots 17
2.3 c o n s t r u c t i o n o f W
λ v s. S p T plots
A first investigation of our sample of 54 lines is done by constructing the Wλ vs. S p T
plot of each line using WTTS spectra taken with SpeX (McClure et al. 2013; McClure 2018, submitted) and the dwarf and giant standard star spectra from the IRTF Spectral Library (Rayner et al. 2009). Measuring the equivalent width in each of the spectra is done with IDL routines from McClure et al. (2013). This method involves a script that dynamically selects continuum regions with the assumption that only the highest fluxes in a given wavelength bin qualify as continuum, and fits a robust polynomial to them. The region is normalized and the area between the flux and the continuum is calculated (see Fig. 15). This is equal to the equivalent width.
Figure 15.: Two plots showing the region around the line at 1.169µm. The left plot shows a broad wavelength range around this line, the spectrum is not normalised. The dot indicates around which wavelength we zoom to obtain the right plot. The right plot is normalized around the line and it shows the fit through the spectral line (blue solid line). From this fit, we calculate the equivalent width.
The spectral types of the WTTS spectra are from McClure (2018, submitted). The resulting plots are presented in App. B. From these plots we make a first selection of lines that appear to meet our selection criteria.
To confirm our selection of lines we use the same technique and the same set of dwarf and giant standard stars, but instead of the WTTS spectra taken with SpeX, we use the spectra we will ultimately analyze: the higher resolution X-shooter spectra from Manara et al. (2017). Since we want to compare these with the same set of dwarf and giant standard stars using the same IDL routines, this involves the convolution of the high-resolution X-shooter spectra (wavelength spacing of ∼0.06nm) to the lower resolution of the dwarf and giant standard stars spectra (taken with SpeX, wavelength spacing of ∼0.21nm). This convolution is done using the Observation module of the Astrolib PySynphot package (Lim et al. 2015). The resulting Wλvs. SpT plots are presented in App. C.
2.4 shaping the models & the data 18
2.4 s h a p i n g t h e m o d e l s & the data
Having produced all the Wλ vs. SpT plots, we are able to select the starspot-sensitive lines
and the two spottiest WTTS in our sample. To constrain the spot fraction and the spot temperature of these WTTS, we compare the line profiles from our data with line profiles from model spectra. As model spectra we use synthetic spectra derived from model atmo-spheres based on the stellar atmosphere code PHOENIX. These atmoatmo-spheres are modeled by defining a metallicity, α-element abundance, temperature and surface gravity (Husser et al. 2013).
2.4.1 Shaping the models
The situation we are modelling is as follows; a stellar photosphere that is subject to two temperatures: the effective temperature that we adopt from the literature, and the temper-ature of the starspots. A fraction of the stellar surface ( f ) has the starspot tempertemper-ature and remaining surface area (1− f ) has the effective temperature. In order to create a single spectrum that resembles this situation, we combine two separate PHOENIX models: one with a temperature that represents our target’s spot temperature and one with a tempera-ture that represents our target’s effective temperatempera-ture. We multiply the flux we obtain from the ‘starspot model’ with f , a number between 0 and 1, which represents the fraction of the stellar surface covered with spots. The flux that we obtain from the ‘effective temperature model’ is multiplied by (1− f ) to account for the remaining part of the stellar surface. For the combined flux we thus find (Fig. 16 shows an example of this):
Fcomb = (1− f) · FTe f f + f· Fspot (2.5)
We generate single temperature PHOENIX models with temperatures varying from 2300K up to the effective temperature of the target source with increments of 100K. Using spot fractions ( f ) of 0.1−1, we combine an ‘effective temperature model’ (i.e. T equal to Te f f
from the literature) with individual ‘starspot models’ that each have a different temperature (varying from 2300K up to the effective temperature of our target). This way, we obtain a two-dimensional grid of two-temperature model spectra, each having its own unique com-bination of Tspot and fspot. Ultimately, we will visualize our fitting results using this grid
(see Fig. 17 for a representation of the grid). Tab. 1 summarizes the ranges between which we vary the model parameters.
2.4 shaping the models & the data 19
Parameter Range Step size Tspot 2300-T
e f f 100
fspot 0.1-1 0.1
Table 1.: Parameter space of the grid of models.
Figure 16.: A plot relating two separate PHOENIX models (blue and orange solid line, Te f f of 3800K and 5000K,
respectively) to its combined, two-temperature model (green line). Here a spot fraction of 80% is used. The edges of the wavelength range of the data is indicated by the dashed lines.
Figure 17.: Visualization of the grid of model atmospheres that is generated. Each?represents a model
atmo-sphere that we compare against the data and for which we, in the end, calculate a goodness-of-fit.
All models in one particular grid have the same effective temperature and log(g)(corresponding to
the values the literature quotes for a specific source), but each model has a unique combination of spot temperature and spot fraction. We will ultimately use the lay-out of this plot to present the fit results.
2.4 shaping the models & the data 20
In order to eventually compare the model spectra to the data, both need to be normalized (continuum flux scaled to 1). Since we are only interested in a couple of line profiles, we cut out an arbitrary region around each of the selected lines and normalize only this region, instead of the whole spectrum. To normalize two-temperature model spectra while conserving the difference in absolute flux of the separate temperature components, we use Eq. 2.6.
Fcomb,norm = f· Fspot+ (1− f) · FTe f f
f · Cspot+ (1− f) · CTe f f (2.6)
Here, the numerator is the combined version of the actual flux (Fspot and FTe f f, scaled with f ) and the denominator is the combined version of the continuum flux (CspotandCTe f f, scaled with f ). We find the continuum fluxes by fitting a cubic polynomial through each spectrum using by hand defined continuum regions (see Fig. 18 for an example), the actual flux (Fspot) is represented by the grey line and the continuum flux (Cspot) is represented by the red line).
The final step we take to shape the synthetic spectra is matching their resolution to the resolution of the data. This is done by using the Observation module of the Astrolib PySynphot package (Lim et al. 2015).
Figure 18.: One of the continuum fits in the synthetic spectra. The by hand defined continuum regions are indi-cated by red dots. This example shows the synthetic spectrum of the PHOENIX model atmosphere with an effective temperature of 3800K around the K line at 1.169 µm.
2.5 fitting models to data 21
2.4.2 Shaping the data
To shape the data in a way that we can compare the models with them, we need to cut out the same arbitrary region around each of the selected lines and normalize them in the same way as we did with the model spectra. Since we do not have the two temperature components as with the models, the normalization of the flux of the data is done by just dividing the actual flux by the continuum:
Fnorm = F
C (2.7)
HereF is the actual flux andCis continuum. Again, the latter is found by fitting a cubic polynomial through by hand defined continuum regions (see Fig. 19). Here we take one extra step, compared to the steps we take with the model spectra. We calculate the error on the normalized flux values that is induced by fitting the continuum (Eq. 2.8) and adopt this as the systematic error on the data. In Eq. 2.8, f it−cont is the deviation of the fit relative to the regions that are defined as continuum and n is the number of continuum points.
rms= s 1 n n
∑
i=0 (f it−cont)2 (2.8) 2.5 f i t t i n g m o d e l s t o d ataAt this point, we have for each line that we have selected: (1) a normalized line profile belonging to a target source and (2) a two-dimensional grid filled with two-temperature model line profiles (all normalized; the spot temperature and spot fraction vary from model to model). To test how well each individual model in our grid agrees with the data, we calculate for each model j the χ2-statistic (Eq. 2.9):
χ2j =
n
∑
i=0
(datai−modeli)2/σi2 (2.9)
χ2j is the χ2-statistic of model j. data−model defines the deviation of the flux of model j
with respect to the data. σiis the rms error induced by fitting the continuum as discussed in
the previous section (calculated using Eq. 2.8). n is the number of data points that lie within the region that is fitted. The wavelength region used for the calculation of the χ2-value is
slightly smaller than the region used to do the continuum fitting, because we want to limit the comparison to the line profile and we do not want to take the region around it into
2.5 fitting models to data 22
Figure 19.: This example shows the spectrum of target PZ1625 around the K line at 1.169µm. Top: One of the continuum fits of the X-shooter data. The by hand defined continuum regions are indicated by red dots. Bottom: The normalised flux, including errors induced by the normalisation.
account (see the pink regions in the final results later on). Having quantified a goodness-of-fit for each model in our grid, we calculate the 1, 2 and 3σ confidence intervals on the spot temperature and the spot fraction. We use that the reduced χ2is defined as:
χ2red= χ
2
d.o. f . (2.10) And the degrees of freedom follow from:
d.o. f .=n−q (2.11) Here n is again the number of data points that are taken into account when fitting and q is the number of free parameters. Since we are fitting for the spot temperature as well as the spot fraction, q=2 in our case.
Finally, since each target naturally has one spot temperature and one spot fraction, we average over the best-fitting models to constrain this. We have one best-fitting model for each spectral line and each best-fitting model gives us a starspot temperature and a starspot fraction. Averaging over these best-fitting temperatures and fractions gives us a mean and a standard deviation for the starspot temperature and the starspot fraction of each target.
3
R E S U LT S
In this section we firstly present the equivalent width vs. spectral type plots of the three lines that we select as starspot-sensitive and we motivate our choice. Then, based on these plots, we select two WTTS targets of which we will compare the data to our models. Finally, we present these results in the form of the three line profiles fitted with our two-temperature models.
3.1 l i n e a n d ta r g e t s e l e c t i o n
App. C shows the equivalent width vs. spectral type plots for all the lines in our sample. After selecting the lines that have the desired dependence of Wλon temperature and surface
gravity, we are left with five lines. The central wavelengths are: 1.0344, 1.1404, 1.169, 1.17751 and 1.2525µm. However, for secondary reasons explained hereafter, the lines at 1.0344 and 1.1404µm are not selected for the final analysis. The line at 1.0344µm is a blend of a calcium line (1.03466µm) and an iron line (1.03437µm), which makes it unclear how the lines individually depend on temperature and log(g). The line that is centered at 1.1404µm is, on closer inspection, not significant, because it does not exceed the noise around it due to a telluric band. One gets an idea of the noise level when looking at the region around 1.14µm in Fig. 9.
The lines at 1.169, 1.17751 and 1.2525µm are selected, based on the dependency of their equivalent width on temperature and surface gravity (see Fig. 20 for their Wλ vs. SpT
plots). For each of these lines, it holds that the equivalent width increases firmly when temperature drops below∼3800K (i.e. these lines are relatively deep if there are regions on the stellar surface with Tlocal < ∼3800K). Furthermore, for the cooler (M-type) stars it holds
that the WTTS lie nicely in between the trends that the giant and the dwarf stars follow and for the hotter (K-type) stars it holds that the WTTS lie above the dwarf trend, as is expected for pre-main-sequence stars that are subject to starspots.
3.1 line and target selection 24
In Fig. 20 we have indicated the targets that we select as spottiest (green and yellow dots). by eye comparing the three plots in Fig. 20, it shows that, on average, these two sources lie the most above the dwarf trend. Tab. 2 shows their stellar parameters.
Figure 20.: Equivalent width versus spectral type plots of the selected lines. Shown are the dwarf stars (blue dots), the giant stars (red dots) and the WTTS (black dots), all with error bars. We have also drawn a parametric fit through the dwarfs (the blue line) and the giants (the red line). Top left: K I line centered at 1.169µm. Top right: K I line centered at 1.17751µm. Bottom: K I line centered at 1.2525µm.
3.2 fit results 25
Object Name SpT Te f f(K) log(g)
PZ99 J160550.5-253313 PZ1625 K1 5097+/- 218 3.81 +/- 0.24 RX J0457.5+2014 RX0420 K1 4841+/- 244 4.51 +/- 0.23
Table 2.: Stellar parameters of the selected sources taken from Manara et al. (2017). In Fig. 20, PZ1625 is colored green and RX0420 is colored yellow.
3.2 f i t r e s u lt s
For each target, we fit three separate line profiles with our two-temperature model spectra. This gives us three spot fractions and temperatures per target. We average over these three values to find a mean spot fraction and spot temperature per target and calculate a corre-sponding standard deviation. Figs. 21-29 show the fits of the best-fitting models (upper panels) and an overview of the goodness-of-fit values of all the models (lower panels), Tab. 3summarizes the best-fitting parameters and Tab. 4 summarizes the parameter values that lie in the 1σ and 3σ confidence intervals for each of the lines.
3.3 f i t t i n g p z 1 6 2 5 d ata
Firstly, we show the results that we find for target PZ1625. According to Manara et al. (2017), this target has a log(g)value of 3.81±0.24. The PHOENIX models that we use to construct our two-temperature models are only available with increments of 0.5 in terms of log(g)(i.e. log(g) =..., 3.0, 3.5, 4.0, ...). Because 3.81 is in between 3.5 and 4.0, we choose to fit the data of PZ1625 with two sets of models, one set of models with log(g) = 3.5 and one set with log(g) =4.0.
3.3.1 Fitting PZ1625 data with models with log(g) =3.5
We fit the three selected line profiles from the spectrum of PZ1625 with two-temperature model atmospheres that have an effective temperature of 5100K and a log(g)of 3.5, while we vary the spot fraction and the spot temperature. We find best-fitting spot temperatures of 4000, 3400 and 4100K for the lines centered at 1.169, 1.17751 and the 1.2525µm, respec-tively. The best-fitting spot fraction for each line is found to be 100%. Combining these results we find for PZ1625 (assuming log(g) =3.5):
3.3 fitting pz1625 data 26
Tspot=3833±309K
fspot =100 ±0 %
Even though we compare only three lines, the standard deviations above give us a quan-titative indication of how internally consistent the results from the different lines are. The upper panels of Figs. 21-23 show how well the average model (Tspot = 3800K, f = 100%;
green, dashed line) agrees with the data of each of the absorption lines separately. For the 1.169µm line and the 1.17751µm line it is within 1σ of the best fitting model. For the 1.2525µm line it is within 2σ. The best-fitting model of each particular line is also shown in the upper panels (blue, solid line).
Looking at the overview of the fit results of the absorption line centered at 1.169µm (Fig. 21, lower panel), it shows that the data is represented similarly well by the models with [Tspot, f ] of [3000K, 100%] and [4000K, 100%]. This could indicate that the surface consists,
besides the effective temperature (5100K), of two extra temperature components: one of
∼3000K and one of∼4000K. However, this is not supported by the lines centered at 1.17751 and 1.2525µm so therefore we reject this as a possibility.
The shape of the confidence intervals belonging to the line centered at 1.17751µm (Fig. 22, lower panel) can be explained by the fact that none of our models correctly fits the depth of both parts of this doublet at the same time. In the upper panel of Fig. 22 it is visible how the model with a spot temperature of 3200K correctly fits the depth of the line at the long wavelength end and how the model with a spot temperature of 3800K correctly fits the depth of the line at the short wavelength end and the region in between the two lines. The models with spot temperatures in between 3200K and 3800K are an interpolation between these two models and thus have a similar goodness-of-fit when one takes into account the full region that is fitted. The best-fitting model (Tspot=3400K) is an example of this.
3.3 fitting pz1625 data 27
PZ1625, line 1.169 µm
Figure 21.: Fit of the best-fitting model and an overview of the results of fitting the data from PZ1625 around
λ =1.169µm with two-temperature model spectra with Te f f = 5000K and log(g) = 3.5. Top: the
data including error bars (black), the best-fitting model (blue, solid line) and the average model (green, dashed line). The pink region indicates which wavelength range is fitted against the data (i.e.
which data points are used to calculate the χ2 statistic). Also included is the model with [3000K,
100%] to show how it compares with the best-fitting model. Bottom: density plot where each dot represents an individual model with a particular spot temperature (x-axis) and spot fraction (y-axis).
The arrow indicates the best-fitting model (see title for χ2redvalue). The 1σ, 2σ and 3σ contours are
given (red, green and blue, respectively) and the color of the dots represents the goodness-of-fit of
3.3 fitting pz1625 data 28
PZ1625, line 1.17751 µm
Figure 22.: Same as Fig. 21. The target is PZ1625, the central wavelength of the line is at λ=1.17751µm. Model
parameters: Te f f =5000K and log(g) =3.5. Top: The models with [3200K, 100%] and [3800K, 100%]
are plotted to show the model spectra that correspond to the edges of the stretched 1σ confidence interval.
3.3 fitting pz1625 data 29
PZ1625, line 1.2525 µm
Figure 23.: Same as Fig. 21. The target is PZ1625, the central wavelength of the line is at λ=1.2525µm. Model
3.3 fitting pz1625 data 30
3.3.2 Fitting PZ1625 data with models with log(g) =4.0
As described previously, we also fit the data of PZ1625 with models that have a log(g) of 4.0, instead of the 3.5 used in the previous analysis. Using the models with log(g) = 4.0, we find best-fitting spot temperatures of 3100K, 3500K and 4200K for the lines centered at 1.169, 1.17751 and 1.2525µm, respectively. For the spot fractions, we again find that each line profile is best represented by a model with a spot fraction of 100%. Averaging over these temperatures and fractions we find the following combined result for PZ1625 (assuming log(g) =4.0):
Tspot=3600±455K
fspot =100 ±0 %
The average model lies outside the 3σ confidence interval for the 1.169µm line, but inside the 1σ confidence interval for the 1.17751µm line and inside the 2σ confidence interval for the 1.2525µm confidence interval.
Comparing with the case where we assume a log(g)of 3.5 for PZ1625, the averaged spot temperature and spot fraction are similar (well within each others standard deviation). The difference is only induced by the result from the 1.169µm line; the result of the other two lines differ only by 100K. Looking at the goodness-of-fit plots of this 1.169µm line (lower panels of Figs. 21 and 24) shows that, in terms of the temperature, even these are very simi-lar: in both cases a relatively cool and a relatively hot temperature region fit the line profile similarly well. It is just that in the log(g) = 4.0 case, the best-fitting model is a model in the cooler temperature region instead of a model in the hotter temperature region, as in the log(g) =3.5 case. All in all, when fitting the PZ1625 data, the difference in results between fitting with models with log(g) =3.5 and models with log(g) =4.0 is insignificant. There-fore we conclude that, regarding the determination of the spot temperature, this method is not heavily dependent on log(g).
Regarding the spot fraction, the log(g) = 3.5 case and the log(g) = 4.0 case produce similar results as well. However, it is interesting to see that for each of the lines the log(g) =
4.0 case allows for slightly lower spot fractions. This is explained by the relation between equivalent width and surface gravity discussed in Chapter 2: for the lines selected in this work, a higher log(g)produces a larger equivalent width and therefore a smaller fraction of the surface is necessary to be covered with cool spots to produce the measured equivalent width.
3.3 fitting pz1625 data 31
PZ1625, line 1.169 µm
Figure 24.: Same as Fig. 21. The target is PZ1625, the central wavelength of the line is at λ=1.169µm. Model
3.3 fitting pz1625 data 32
PZ1625, line 1.17751 µm
Figure 25.: Same as Fig. 22. The target is PZ1625, the central wavelength of the line is at λ=1.17751µm. Model
3.3 fitting pz1625 data 33
PZ1625, line 1.2525 µm
Figure 26.: Same as Fig. 21. The target is PZ1625, the central wavelength of the line is at λ=1.2525µm. Model
3.4 fitting rx0420 data 34
3.4 f i t t i n g r x 0 4 2 0 d ata
We fit the three line profiles from the spectrum of RX0420 with two-temperature models with an effective temperature of 4800K and a log(g)of 4.5 while we vary the spot temper-ature and the spot fraction. For the lines at 1.169, 1.17751 and 1.2525µm we find [Tspot, f ]
values of [3300K, 100%], [2600K, 70%] and [3000K, 80%], respectively. Combining these results into a single spot temperature and spot fraction:
Tspot=2967±287K
fspot =83 ±12 %
The average model lies outside the 3σ confidence interval for the 1.169µm line as well as for the 1.17751µm line, but inside the 1σ confidence interval for the 1.2525µm confidence interval.
Interestingly, when one qualitatively compares the goodness-of-fit overviews of the three different lines (lower panels of Figs. 27-29), it is evident that all show the same ‘trend’ for which models represent the data reasonably well: from about [Tspot, f ] = [2500K, 60%]
increasing linearly up to [3400K, 100%]. This can be interpreted as a degeneracy that is present in spot temperature and spot fraction when these parameters are varied to repro-duce an absorption line. Having a solution, one can increase (decrease) the spot tempera-ture and increase (decrease) the spot fraction to find another, similarly well fitting solution. This property of the parameter space did not occur with the data from PZ1625, but this might be related to the fact that in the latter case all the best-fitting models have a spot fraction of 100%.
Important to note, is that the best fit of the 1.169µm line does not represent the data very well (χ2redvalue of 38.8; Fig. 27), compared to the other fits. Therefore one could argue that the Tspotand f values resulting from this fit are to a lesser extent representative for the real
3.4 fitting rx0420 data 35
RX0420, line 1.169 µm
Figure 27.: Same as Fig. 21. The target is RX0420, the central wavelength of the line is at λ=1.169µm. Model
parameters: Te f f =4800K and log(g) =4.5. Top: Also plotted is the other end of the ‘linear trend’
that is visible in all the overview plots: the model with parameters [2700K, 70%]. It shows how, by eye, the deviation from the best-fitting model ([3300K, 100%]) is rather minimal.
3.4 fitting rx0420 data 36
RX0420, line 1.17751 µm
Figure 28.: Same as Fig. 21. The target is RX0420, the central wavelength of the line is at λ=1.17751µm. Model
3.4 fitting rx0420 data 37
RX0420, line 1.2525 µm
Figure 29.: Same as Fig. 21. The target is RX0420, the central wavelength of the line is at λ=1.2525µm. Model
parameters: Te f f =4800K and log(g) =4.5. Top: Note how the best-fitting model and the average
3.4 fitting rx0420 data 38
Best-fitting models
Lines Combined result
Objectlog(g) Param 1.169µm 1.17751µm 1.2525µm µ σ
PZ16253.5 χ2red 7.17 1.37 10.8 Tspot 4000K 3400K 4100K 3833K 309K f 100% 100% 100% 100% 0% PZ16254.0 χ2red 6.16 1.00 12.0 Tspot 3100K 3500K 4200K 3600K 455K f 100% 100% 100% 100% 0% RX0420 χ 2 red 38.8 17.1 13.9 Tspot 3300K 2600K 3000K 2967K 287K f 100% 70% 80% 83% 12%
3.4 fitting rx0420 data 39
1
σ
and
3
σ
confidence
inter
v
als
Lines 1. 169 µ m 1. 17751 µ m 1. 2525 µ m O b je ctlo g ( g ) Param χ 2 red 1 σ 3 σ χ 2 red 1 σ 3 σ χ 2 red 1 σ 3 σ P Z 1625 3.5 Tsp ot 7. 17 2900 -3000 K 3700 -4300 K 2400 -2600 K 2800 -4500 K 1. 37 3200 -3900 K 2600 -4000 K 10 . 8 3600 K 4000 -4200 K 3500 -4300 K f 100% 90−
100% 100% 90−
100% 100% 100% P Z 1625 4.0 Tsp ot 6. 16 3100 -3200 K 4000 -4300 K 2700 -3400 K 3700 -4500 K 1. 00 3000 -3200 K 3400 -4100 K 2700 -4200 K 12 . 0 3700 -4300 K 3600 -4400 K f 100% 90−
100% 90−
100% 90−
100% 100% 90−
100% R X 0420 Tsp ot 38 . 8 3300 K 3300 K 17 . 1 2600 -2700 K 2500 -2700 K 2900 K 13 . 9 2700 -3100 K 3200 K 2500 -3300 K f 100% 100% 70% 70% 80% 70−
80% 90% 60−
90% T able 4. : An indication of the goodness-of-fit (χ 2 red ) and the found spot temperatur es in the 1 σ and 3 σ confidence inter v als. The results regar ding PZ 1625 ar e divided o v er tw o ro ws since w e ha v e fitted the data fr om this sour ce with PHOENIX models with a lo g ( g ) of 3. 5 and 4. 0.4
D I S C U S S I O N
By averaging over the fit results from three line profiles, we have constrained one starspot fraction and one corresponding temperature for each of our two targets, PZ1625 and RX0420. In this section we firstly discuss the interpretation and the implications of our findings. Secondly, we review the method that we use and discuss its advantages and shortcomings. Finally, we consider how this method can be applied to classical T Tauri stars, the hosts of protoplanetary disks.
4.1 t h e c o n s t r a i n e d s p o t f r a c t i o n s a n d t e m p e r at u r e s
For PZ1625 we find a spot fraction of 100%, regardless of the surface gravity we assign to the models. When we fit with models that have log(g) =3.5, we find a spot temperature of 3833±309K; when we fit with models that have log(g) = 4.0, we find a spot temperature of 3600±455K. As we discussed in Section 3.3.2, this (small) difference is only induced by the line centered at 1.169µm. This line gives Tspot = 3100K in the log(g) = 3.5 case and
Tspot=4000K in the log(g) =4.0 case. In both cases (log g=3.5 and log g =4.0) the lower
Tspot and the higher Tspot both produce acceptable fits. Therefore we qualify the results
as not significantly different and therefore we will not differentiate between the two cases when interpreting the results. In this part we adopt a spot temperature of ∼3600-3800K and a spot fraction of 100% for PZ1625.
For RX0420 we find a spot fraction of 83±12% and a spot temperature of 2967±287K.
4.1.1 Discussion of the found starspot fractions
The starspot coverage fractions found in this work seem artificially high: fspotPZ = 100%
and fspotRX =83±12%. However, a high spot fraction is not unknown. Gully-Santiago et al.
(2017) have found that 80% of the surface of the WTTS LkCa 4 is covered with cool spots and
4.1 the constrained spot fractions and temperatures 41
they argue that this is not an extreme case. This is supported by earlier studies of Grankin that find large spot fraction too. Maximum spot coverage fractions of 67% (Grankin 1999) and 90% (Grankin 1998) are quoted. The 71−95% spot fraction that we find for RX0420 therefore does not contradict the literature per se.
The spot fraction of 100% that we find for PZ1625 is more suspicious, because it overlaps with the edge of our grid of models. This may be a consequence of the spacing of our grid in terms of spot fraction (we use increments of 10%). It may be the case that if one uses an increment of 1% for the spot fraction, the best-fitting model has a spot fraction of somewhere in between 90−100%. Regardless, this is still an extremely high spot fraction.
A credible interpretation for high spot fractions is that the star in question has an inclined axis. A stellar axis that is inclined into our line of sight would contribute to finding a spot fraction that is higher than the actual surface coverage of the starspots. Carroll et al. (2012) have mapped the surface temperature for WTTS V410 Tau (see Fig. 30, bottom) and they show that the cool areas of the surface are situated at the poles. If one would be viewing a TTS pole-on, a high spot fraction is expected. Due to limb darkening effects of the hotter regions (that are then situated at the equator) the found spot fraction will be even higher. Donati et al. (2014) have shown for LkCa 4 (the WTTS source that Gully-Santiago et al. (2017) investigated) that we view this source pole-on and they have constructed a brightness map (see Fig. 30, top). This inclination effect is thus sufficient to explain the spot fraction of 83±12% that we find for RX0420. However, it is implausible that the inclination effect alone is sufficient to explain the spot fraction of 100% that we find for PZ1625. When we review our method (Section 4.2.2) we will come back to this.
4.1.2 Discussion of the found starspot temperatures
The starspots temperatures that we find,∼3600-3800K for PZ1625 and∼3000K for RX0420, are in agreement with the known relation between effective temperature and starspot tem-perature: ∆T = 200-2000K, where the ∆T is generally bigger for G-type and early K-type stars than for M-type stars (e.g. Berdyugina (2005). PZ1625 and RX0420 have spectral type K1 (Manara et al. 2017) and temperature differences of: Te f f −Tspot = ∼1400K for PZ1625
and Te f f −Tspot= ∼1800K for RX0420.
As discussed in the introduction, the radial location of planet formation is related to the presence of optically thick gas in the innermost regions of protoplanetary disks, due to shadowing effects. It is thought that when investigating whether this optically thick gas is
4.1 the constrained spot fractions and temperatures 42
Figure 30.: Top: figure from Donati et al. (2014) showing the brightness distribution on the surface of LkCa
4(relative to the quiet photosphere) as viewed from Earth (pole-on). Bottom: figure from Carroll
et al. (2012) showing the temperature distribution on the surface of V410 Tau at different rotational phases φ.
present or not, one has to be careful not to confuse its signatures with that of cooler patches on the stellar surface (Wolk & Walter 1996), i.e. starspots. This confusion is lurking if one considers an SED and if the temperature of these starspots and the temperature of this optically thick gas are similar. In this work, we have investigated weak-lined T Tauri stars (generally thought to have no disk) as opposed to classical T Tauri stars (generally thought to be hosting a PPD), but since these different types of stars have very similar physical properties (Stempels & Piskunov 2003; Cieza et al. 2005), we believe it is worthy to apply the quantitative results we have found for WTTS to CTTS.
We find starspot temperatures of ∼3600-3800K for PZ1625 and ∼3000K for RX0420. Re-lating this to the temperature associated with the innermost regions of the disk (about 2000-6000K (Muzerolle et al. 2004), it is evident that these two sources of light are easily confused if one only takes an SED into account. In other words, to determine whether this optically thick gas is present or not, one needs other tracing methods than an SED or one needs to quantify the contribution of starspots to the total SED. Another method to trace the gas is, for example, NIR spectro-interferometry (Kraus et al. 2008, 2009; Benisty et al. 2010). This work, however, contributes to quantifying the contribution of starspots to the
4.2 reviewing the method 43
total SED, since we are testing a method that determines a starspot temperature and a cov-erage fraction. These parameters (temperature and area) combine to a ‘starspot SED’ when one multiplies Planck’s formula (Eq. 1.1, depends on T) by the area that the starspots cover. Such a starspot SED is another piece of the puzzle in defining the NIR excess emission that we find in light from star-disk systems (e.g. such a starspot SED would be as one of the dotted lines in Fig. 7).
4.2 r e v i e w i n g t h e m e t h o d
In this part we firstly discuss the internal consistence of our results. Secondly, we review the method: how we use the Wλ vs. SpT plots to select the starspot-sensitive lines and the
spottiest WTTS targets, followed by how we fit the line profiles to the PHOENIX models.
4.2.1 Consistency of the results over the different lines
An important point regarding the evaluation of the method is the internal consistence of the results. Each target has one true starspot fraction and one true starspot temperature which we are trying to determine using the line profiles of three different spectral lines. A proper method should give similar results across the different lines. In this work this is the case.
Looking at the results for PZ1625 in the case where we fit the data with models that have log(g) =3.5, it holds that the model that corresponds to the average spot fraction and temperature (100% and 3800K) lies within a 1σ confidence interval of the best-fitting models when looking at the lines centered at 1.169 and 1.17751µm. It lies within a 2σ confidence interval of the line centered at 1.2525µm.
Regarding PZ1625 in the case where we fit the data with models that have log(g) = 4.0, the average model (100% and 3600K) lies outside of the 3σ confidence interval for the line at 1.169µm, but inside the 1σ confidence intervals for the lines at 1.1771 and 1.2525µm. However, as previously discussed, the line profile of the line at 1.169µm is similarly well produced by the second-best fit: f =100% and Tspot =4200K. This second-best fit has a χ2red
value of 6.26 as opposed to the 6.16 that corresponds to the best-fitting model ( f = 100% and Tspot=3100K). If we consider the (almost equally likely) scenario where the line profile
of the 1.169µm line would be best-fitted by the model with f =100% and Tspot=4200K, we
would obtain an average model with parameters f = 100% and Tspot = 3967±330K. The
spot fraction and the spot temperature of this average model lies inside the 1σ confidence interval for each of the lines.
4.2 reviewing the method 44
For RX0420 the results across the lines seem not very consistent. The average model ( f = 80% and Tspot = 3000K) lies outside the 3σ confidence interval for the lines at 1.169
and 1.17751µm. However, the fact that the earlier discussed ‘trend’ is visible in each of the goodness-of-fit plots (lower panels of Figs. 27-29) is a qualitative form of internal consistence and it indicates that these three lines trace the same phenomenon.
4.2.2 Temperature dependence of W
λ
In this work, an important criterion for selecting spectral lines is how their equivalent width depends on temperature, according to the Wλvs. SpT plots. The aspect we valued the most
is a steep temperature dependence around the expected starspot temperature. One thing that should be kept in mind, however, is that if a spectral line has no temperature depen-dence around the effective temperature of a star, this method will only give information about the starspot temperature and not about the starspot fraction. This, because the spec-tral line is in this case only formed in the region with Tlocal = Tspot. It might be the case that
the three lines selected in this work, 1.169, 1.17751 and 1.2525µm, all are insensitive to the effective temperature of target PZ1625 (Te f f =5100K). This idea is somewhat supported by the Wλ vs. SpT plots (see Fig. 20), but just this information is not decisive at this point. To
understand what the temperature distribution on the surface is and to crosscheck the re-sults from this study one could use spectropolarimetric data to construct a brightness map such as that from Donati et al. (2014) (Fig. 30, top) or such as figures 4 & 5 from Nicholson et al. (2018).
4.2.3 Importance of the ‘correct’ spectral type and Te f f
In the Wλ vs. SpT plots, the assigned spectral type of a source determines its horizontal
placing. If a star is assigned the wrong spectral type, the Wλ vs. SpT plots give the idea
that the star is more or less heavily spotted than it in reality is. This could lead to not choosing the heaviest spotted target from a sample. The spectral types adopted in this work are derived from optical spectra (Manara et al. 2017). It is known that spectral types derived from optical spectra have the tendency to be too early, due to the ‘filling in’ (or veiling) of optical lines as a consequence of continuous emission origination from accretion shocks (Bouvier & Appenzeller 1992; Mora et al. 2001; Vacca & Sandell 2011). If a source is assigned a spectral type that is earlier than its true spectral type, this results in a higher apparent spot fraction. Fig. 31 explains why this is the case. The expected spot temperature
