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In Search of Temperature Minimum at Radio Wavelengths

Mon, 01 May 2023 00:00:00 +0000

Introduction

Our ability to infer the physical characteristics of stellar atmospheres largely depends on our understanding of the emissions from stars at sub-millimeter to centimeter wavelengths. Unfortunately, the technical difficulty in observing main sequence stars at these wavelengths has delayed research in this area. However, with the development of new infrastructures such as VLA and ALMA, we have been able to explore stars other than our sun for the first time (Liseau et al. 2013, 2015, 2016 ¹ ² ³). This has allowed us to develop a methodology to estimate the temperature structure in the upper atmosphere (Tapia-Vázquez & De la Luz, 2020 ⁴). As a result, we have been able to establish that in solar-type stars the primary structure is similar, that is to say, the atmosphere is made up of five layers:

Phostosphere
Temperature Minimum Region
Chromosphere
Transition Region
Corona

Photosphere

The photosphere can be defined as that portion of the stellar atmosphere visible in optical light where the continuum is optically thick, or nearly so, and where the temperature decreases as height increase because non-radiative heat sources are not strong enough to significantly modify the balance of radiative and convective energy input against radiative losses to space. This layer is studied with visible and infrared light.

Sun’s photosphere with sunspots. ®ESO, 2004

Chromosphere

The chromosphere is an atmospheric region that begins at an atmospheric layer where there is a measurable amount of local heating in addition to radiative and convective heat transport from below (Linsky, 2017 ⁵). This layer has been studied mainly in the EUV and UV. However, due to its physical nature, it is possible to study it at wavelengths ranging from the far infrared to millimeters.

Sun’s chromosphere. ®SOHO, 2019

Transition Region

The transition region is the atmospheric layer where we can observe a sudden increase in temperature within a few kilometers. Study of this region is mainly focused on the EUV using space probes such as SOHO.

Sun’s transition region. ®SOHO, 2019

Corona

The extended outer atmosphere of the Sun is called the corona. It has a temperature of millions of degrees and is where the solar wind originates. This atmospheric layer is mainly studied using x-rays and radio frequencies below 1 GHz.

Sun’s corona . ®P. Horálek/ESO, 2019

What about the Temperature Minimum Region?

At a certain height in the atmosphere, the temperature starts to increase as altitude increases due to a combination of reduced radiative cooling and nonradiative heating as the density decreases. This marks the lower boundary of the chromosphere, which begins at the minimum temperature ($T_{min}$) located at the top of the photosphere and extends upward as temperature increases. The upper photosphere has relatively high densities, resulting in efficient radiative cooling, whereas the nonradiative heating rate in the lower-density layers of the chromosphere is significantly greater. In this context we can say that the minimum temperature region is the layer where the temperature drops to its lowest point. It occurs approximately 500 km (310 miles) above the photosphere. The temperature in this region decreases in the case of the sun from about 5,500 K in the photosphere to around 3,800 K ⁵. This minimum temperature has been observed since the 1950s using observations of the sun at wavelengths between the far infrared and submillimeter. Due to the limitations of the instruments, it was not possible to carry out a census of solar-type stars. It was not until 2013 that Liseau et al. (2013) ¹, using the Herschel space telescope, was able to measure the minimum temperature in a star other than the Sun for the first time. Subsequently, White et al. 2020 ⁶ discovered a minimum temperature in an F-type star. More recently, White et al. 2021 ⁷ found that A-type stars like Altair can also find the temperature minimum but at millimeter wavelengths.

How can we say that we see the minimum of temperature?

In order to say that we are observing the minimum temperature, we must make observations of the star’s atmosphere at different wavelengths. This does allow us to create an SED and analyze its behavior. In figure 5 we can see that by using various facilities we can reconstruct the emission of Altair’s atmosphere. We can see that at around 1.1 mm a minimum temperature is reached and at longer wavelengths it begins to increase. We can then say that we have found the minimum temperature.

Altair’s SED. (White et al. 2021)

Why is the minimum temperature important?

In our most recent work (Tapia-Vázquez et al. submitted) we have used observations at wavelengths ranging from the far infrared to the millimeter to build models of the temperature structure that have revealed the existence of temperature minima in stars with effective temperatures smaller or equal than 7,500 K. Those models allowed us to establish a relationship between the minimum temperature and the effective temperature of the star. We also established a relationship between the frequency at which the temperature minimum is observed and the effective temperature of the star. However, for stars with effective temperatures around 10,000 K, observations reveal that the temperature structure changes their morphology. White et al. (2020) ⁶ show that for Sirius A, there is no warming in the atmosphere; on the contrary, the temperature drops as we move away from the photosphere.

(Tapia-Vázquez et al. submitted)

Remember that for stars with a mass greater than 1.5 ms the internal structure changes. The core becomes convective and the transport of energy towards the upper layers is carried out by radiative transfer. Not having a convective zone near the photosphere, this type of star lacks strong magnetic fields, so layers such as the chromosphere and the corona do not exist. Therefore, the existence of this layer has a direct impact on the dynamics and heating of the outer solar atmosphere. Nevertheless, the physical mechanism supporting the existence of this region remains unknown but can be related to internal structure.

How can the ngVLA help to understand this region?

Günther et al.(2022)⁸ proposes that the internal structure changes when the effective temperature of the star is around 8,100 K. If the temperature minimum is related to the internal structure of the star, then we should be able to see the change in temperature structure around this effective temperature. Due to their unique capabilities, such as high sensitivity and high angular resolution, both ngVLA and SKA will enable us to census stars whose effective temperatures are between 7,500 K and 10,000 K. Combining these observations with other wavelengths obtained with facilities such as ALMA or the James Webb telescope, we will be able to understand what physical mechanism allows minimum temperature and what its relationship with the dynamics of the atmosphere is.

This image shows the expected flux for an A4V star. Using the ngVLA calculator, one hour of telescope observation reveals stars within a radius of 100 pc at a frequency of 93 GHz. However, if we want to observe at lower frequencies simultaneously, we must limit the radius to around 30 pc.

Acknowledgement

This work was supported by the Ciencia Básica (254497) CONACyT Fellowship. The author is grateful for the PAEP-UNAM fellowship.

References

Liseau, R., Montesinos, B., Olofsson, G., et al. 2013, A&A, 549, L7 Doi ↩︎
Liseau, R., Vlemmings, W., Bayo, A., et al. 2015, A&A, 573 Doi ↩︎
Liseau, R., De la Luz, V., O’Gorman, E., et al. 2016, A&A, 594, A109 Doi ↩︎
Tapia-Vázquez, F., & De la Luz, V. 2020, ApJS, 246, 5 Doi ↩︎
Linsky, J.~L. 2017, ARA&A, 55, 159. Doi ↩︎
White, J. A., Tapia-Vázquez, F., Hughes, A. G., et al. 2020, ApJ, 894, 76 Doi ↩︎
White, J.~A., Tapia-V{'a}zquez, F., Hughes, A.~G., et al. 2021, APJL, 912, L5 Doi ↩︎
Günther, H.~M., Melis, C., Robrade, J., et al. 2022, AJ, 164, 8 Doi ↩︎

New Insights into Stellar Atmospheres at Millimeter, Sub-millimeter, and Infrared wavelengths

Wed, 15 Jun 2022 00:00:00 +0000

Introduction

The radio spectra of main-sequence stars remain largely unconstrained due to the lack of observational data to inform stellar atmosphere models. As such, the dominant emission mechanisms at long wavelengths, how they vary with spectral type, and how much they contribute to the expected brightness at a given radio wavelength are still relatively unknown for most spectral types. The particular characteristics of the emission at millimeter, submillimeter, and infrared wavelengths in the solar chromosphere allow us to estimate their temperature and density using indirect methodologies like semiempirical models (Vernazza et al. 1981; Fontenla et al. 1993; Avrett & Loeser 2008). These models are an important tool for a wide set of studies, e.g., solar and stellar chromospheres (Loukitcheva et al. 2004;Linsky 2017), temperature minimum (Liseau et al. 2013; De la Luz et al. 2014), solar flares (Machado et al. 1980; Trottet et al. 2015), and Sun component features (Fontenla et al. 2006).

Semiempirical models predict values close to photospheric temperatures that decrease until a minimum, then increase dramatically until the coronae (Vernazza et al. 1981; Avrett & Loeser 2008. This layer is known as the solar chromosphere. The chromosphere remains observable by different spectral ranges that include ultraviolet in the continuum and line emission, in the visible (mainly Hα), and in the millimeter, submillimeter, and infrared wavelengths. This last wavelength range becomes more important as improvements in the sensitivity and space resolution in modern radio telescopes are made (Nakajima et al. 1995; Kudaka et al. 2015; Wedemeyer et al. 2016).

In this work, we present a new relation between temperature minimum and effective temperaature of the star. We use the models obtained with KINICH PAKAL (Tapia-vázquez & De la Luz, 2020) to construct a grid of semiempirical models. Addicionally, we construct a grid of observed brightness temperature. The precise location of the temperature minimum depends on the detailed structure of the atmosphere and can, with no convincing theory at hand, only be determined from direct measurement. It is in this region where non-radiative energy is deposited, and its physics is of strong general interest (Liseau et al. 2013).

Methods

Our work is based in previous models of main sequence stars obtained from FIR to mm wavelenghts observations (White et al. 2021; White et al. 2020; Tapia-Vázquez & De la luz, 2020).

Observations

Observations at frequencies greater than 1000 GHz were obtained from Herschel data archival. Beetween 1000 GHz and 90 GHz observations came from ALMA and NOEMA. For low frequencies like 33 GHz and 17 GHz, observations were performand with VLA and ATCA.

In literature we can (Villadsen et al. 2014, Liseau et al. 2016, White et al. 2018, Rodríguez et al. 2019, White et al. 2020, White et al. 2021)

KINICH PAKAL

At far-infrared/millimeter wavelengths, stellar emission in main sequence stars is dominated by optically thick free–free radiation (Dulk 1985; Güdel 2002). The flux is proportional to the plasma temperature ($T_{R}$) at a given wavelength and can be used to probe the temperature structure as a function of height above the photosphere. The stellar spectrum can therefore be used to build a model of the thermal structure of the chromosphere. These models were generated using the KINICH-PAKAL code (Tapia-Vázquez & De la Luz 2020). This code iteratively modifies the radial temperature and hydrogen density profiles, the ionization balance, and the opacity of a base model using the Levenberg–Marquardt algorithm to adjust the synthetic spectrum to the ALMA data presented here and the Herschel/PACS data for γ Lep (Montesinos et al. 2016). In the atmosphere, the chromosphere has a higher temperature than the photosphere leading to a strong deviation from radiative equilibrium. Therefore, we do not assume that ionization-excitation and radiative transfer are in local thermal equilibrium. For our models, a semiempirical solar model (model C7 from Avrett & Loeser 2008) in hydrostatic equilibrium was adopted as the starting point. This can be taken as an average of the most commonly used solar models (Vernazza et al. 1981; Fontenla et al. 1993; Loukitcheva et al. 2004). For stars with a higher effective temperature than the Sun, this model serves as an initial condition.

Results

Observational grid

In figure 1, we show the observation grid obtained from the interpolation of 11 stars found in the literature. While the effective temperature is increasing, the frequency at which the minimum temperature is located is approaching the low frequencies. This behavior can be characterized using the equation

$$ \begin{equation} \label{eq:obs_grid} \nu_{T_{B_{min}}}(T_{eff}) = a log(b T_{eff}) + c \end{equation} $$

Where

$\nu$ is the observed frequency
$T_{B_{min}}$ is minimum brightness temperature
$T_{eff}$ is the efective temperature of the start
a=-6.81$x10^{3}$
b=1.41$x10^{-3}$
c=1.65$x10^{4}$

Figure 1. Obseervationaal grid.

Semiempirical model grid

Figure 2 shows the sempiempirical grid model. This grid is composed of the published semipyrical models for 7 stars, including the sun. For the missing stars it was not possible to obtain a reliable model, mainly due to the lack of observations. As an example, we have $\tau$ Cet which was observed by Villadessen et al., (2014). On that occasion, it was detected at 33 GHz and not at 15 GHz, so building a model is not feasible, since we would only be modeling a very small region of the atmosphere.

$$ \begin{equation} \label{eq:minimo} h_{T_{r,min}}(T_{eff}) = aT_{eff}+ bT_{eff}^2 +c \end{equation} $$

Where

h is the height at which the minimum radial temperature iss located
$T_{r, min}$ is minimum radial temperature
$T_{eff}$ is the efective temperature of the start
a=-3.33
b=3.11x10$^{-4}$
c=9.37x10$^3$

Figure 2. SemiemModel grid

Conclusion

We found a relationship between the radial temperature minimum and the effective temperature of the star.
The frequency at which the observed temperature minimum is found varies with the effective temperature of the star.

Acknowledgement

This work was supported by the Ciencia Básica (254497, bkhbvjkhfb) CONACyT Fellowship. The author thank to PAEP-UNAM fellowship.

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