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date: 28 May 2020

# Solar Photosphere

## Summary and Keywords

The Sun is a G2V star with an effective temperature of 5780 K. As the nearest star to Earth and the biggest object in the solar system, it serves as a reference for fundamental astronomical parameters such as stellar mass, luminosity, and elemental abundances. It also serves as a plasma physics laboratory. A great deal of researchers’ understanding of the Sun comes from its electromagnetic radiation, which is close to that of a blackbody whose emission peaks at a wavelength of around 5,000 Å and extends into the near UV and infrared. The bulk of this radiation escapes from the solar surface, from a layer that is a mere 100 km thick. This surface from where the photons escape into the heliosphere and beyond, together with the roughly 400–500 km thick atmospheric layer immediately above it (where the temperature falls off monotonically with distance from the Sun), is termed the solar photosphere.

Observations of the solar photosphere have led to some important discoveries in modern-day astronomy and astrophysics. At low spatial resolution, the photosphere is nearly featureless. However, naked-eye solar observations, the oldest of which can plausibly be dated back to 800 bc, have shown there to be occasional blemishes or spots. Systematic observations made with telescopes from the early 1600s onward have provided further information on the evolution of these sunspots whose typical spatial extent is 10,000 km at the solar surface. Continued observations of these sunspots later revealed that they increase and decrease in number with a period of about 11 years and that they actually are a manifestation of the Sun’s magnetic field (representing the first observation of an extraterrestrial magnetic field). This established the presence of magnetic cycles on the Sun responsible for the observed cyclic behavior of solar activity. Such magnetic activity is now known to exist in other stars as well.

Superimposed on the solar blackbody spectrum are numerous spectral lines from different atomic species that arise due to the absorption of photons at certain wavelengths by those atoms, in the cooler photospheric plasma overlying the solar surface. These spectral lines provide diagnostics of the properties and dynamics of the underlying plasma (e.g., the granulation due to convection and the solar p-mode oscillations) and of the solar magnetic field. Since the early 20th century, researchers have used these spectral lines and the accompanying polarimetric signals to decode the physics of the solar photosphere and its magnetic structures, including sunspots. Modern observations with high spatial (0.15 arcsec, corresponding to 100 km on the solar surface) and spectral (10 mÅ) resolutions reveal a tapestry of the magnetized plasma with structures down to tens of kilometers at the photosphere (three orders of magnitude smaller than sunspots). Such observations, combined with advanced numerical models, provide further clues to the very important role of the magnetic field in solar and stellar structures and the variability in their brightness. Being the lowest directly observable layer of the Sun, the photosphere is also a window into the solar interior by means of helioseismology, which makes use of the p-mode oscillations. Furthermore, being the lowest layer of the solar atmosphere, the photosphere provides key insights into another long-standing mystery, that above the temperature-minimum

(~500 km above the surface at ~4000 K), the plasma in the extended corona (invisible to the naked eye except during a total solar eclipse) is heated to temperatures up to 1,000 times higher than at the visible surface. The physics of the solar photosphere is thus central to the understanding of many solar and stellar phenomena.

# 1. Introduction

Continued advances in observing techniques since the invention of the telescope in the early 1600s help scientists to probe the Sun in ever greater detail. To date, the Sun is the only star that can be spatially resolved down to a scale of 100 km, with a temporal resolution or cadence of the order of 1 s. These unprecedented modern-day observations along with the knowledge on the cyclic and secular evolution of solar activity that has accumulated over several millennia are valuable to understanding the evolution of billions of stars in our Galaxy and beyond. The ability to scrutinize the gaseous body that is our Sun enables us to build a theoretical framework and computer-based models to extract its physics, and by extrapolation, the physics of other stars (an overview of solar physics is provided in Priest, 2019a).1

Perhaps the most widely used and yet simple of all methods to observe the Sun is through its electromagnetic radiation. The energy that is generated by nuclear fusion in the core of the Sun is transported outward, first through radiation and then through convective processes (i.e., the bulk motion of the gas). This energy then escapes the Sun as electromagnetic radiation through its 100 km thick surface. Beneath this surface the Sun is completely opaque to radiation. Thus the photosphere represents the outer layer of the solar gas, which changes from completely opaque to being almost transparent. Through this layer almost all the photons escape into the heliosphere. The energy spectrum of these photons escaping the Sun has the same integral as that of a blackbody radiating at a temperature of 5780 K, defined as the effective temperature of the Sun. However, prior to escaping the Sun, photons at certain energies do interact with the atoms that constitute photospheric gas, causing absorption of radiation at those wavelengths. This blackbody-like radiation and the absorption features superimposed on it carry the information on the apparent characteristics of the solar surface and its structures.

The most prominent structures visible at the photosphere are produced by convection and the magnetic field. In the Sun, the convection zone extends from 0.7$R⊙$ to the surface (Stix, 2002). At the base of this convection zone, the ionized gas (plasma) heated from the radiation below is transported to the surface. It cools near the photosphere (due to the radiation escaping the Sun) and sinks back to the depths of the convection zone, where it is reheated. This process of convection repeats itself, transporting the energy from the hotter interior of the Sun to its cooler surface. This continuous convective churning or overturning creates eddies or cells of plasma moving through the convection zone. Near photospheric layers, the cells of plasma form structures with a range of spatial scales from supergranules (~30,000 km) to granules (~1,000 km) (much like the bubbles forming on the surface of boiling water). These convective motions play a fundamental role in maintaining the magnetic field of the Sun through the dynamo action (Cameron, 2019). An intensity map of the solar photosphere with various features (e.g., sunspot, pores, granules) is displayed in Figure 1. On larger spatial scales compared to granules, the magnetic field is structured into darker pores or sunspots (darker compared to the granular brightness). Under appropriate conditions, sunspots are actually visible to the unaided eye as dark specks on the solar disk (see Stephenson & Willis, 1999, for a discussion on the earliest drawings of sunspots). On much smaller scales in the regions between granules (intergranular lanes), the magnetic field is seen as bright features when observed in narrow wavelength bands in the near-ultraviolet to near-infrared range of the electromagnetic spectrum (Solanki, 1993). Overall, the magnetic field evolves on timescales of minutes to years that influence the short-term and long-term solar activity, with crucial implications for planet Earth (Haigh, 2007; Usoskin, 2017).

The information of convective motions and the magnetic field is imprinted on the radiation emergent from the photosphere. For instance, the up- and downward motions of granules shift the position of the absorption features (line profiles) in wavelength. Therefore, the oscillations and flows generated by the convection can be studied by analyzing the shifts in the line profile produced by the Doppler effect. By employing the techniques of helioseismology, the structure of the convection zone and the systematic plasma flows through it can be inferred from p-mode oscillations detected in the photosphere (Gizon & Birch, 2005; Gizon & Käpylä, 2019). Furthermore, the magnetic field causes splitting or broadening of spectral-line profiles and polarizes the radiation. Using the Zeeman effect, for example, the properties of the solar magnetic field at the photosphere can be inferred. Measuring the photospheric magnetic field is crucial to understanding the solar activity from timescales of minutes to years.

The theoretical formulations required to decode photospheric observations are also well developed. The theory of radiative transfer deals with the interaction of light with atoms in the presence of magnetic fields (Stenflo, 1994; del Toro Iniesta, 2003; Hubeny & Mihalas, 2014). The equations of magnetohydrodynamics (MHD) (Priest, 2019b) are employed to study the evolution of convection and magnetic fields (magnetoconvection) in the Sun and its photosphere (Nordlund et al., 2009; Stein, 2012). Computationally intensive simulations in general have achieved a measure of realism and are well tested against the observations. New data continue to provide more physical constraints and help refine the models by revealing any missing physics in them.

This article presents a general introduction to the solar photosphere. It focuses on the topics of radiation (sect. 2), convection (sect. 3.1), and magnetism (sect. 3.2), the three branches of photospheric research. How these three branches are combined to develop state-of-the-art realistic numerical models of the solar photosphere is then discussed (sect. 3.3). Finally, some of the open questions in the field are outlined that will be addressed with the upcoming and next-generation solar telescopes (sect. 4).

Figure 1. Various features of the solar photosphere observed in the G-band at 4,305 Å with the Solar Optical Telescope onboard Hinode on May 2. 2007 (Kosugi et al., 2007; Tsuneta et al., 2008). The prominent dark feature is a sunspot. The large dark area in the map is the umbra of the sunspot, which is surrounded by fibril-like penumbra. The other prominent structures include the solar granulation and smaller darker regions known as pores. The solar granulation is divided by the slightly darker intergranular lanes. In some places, due to the concentration of small-scale magnetic field, the intergranular lanes appear brighter (see Figure 5 for a closer look at these smaller magnetic features).

Before escaping into the heliosphere, the radiation emitted from the hot core of the Sun passes through the stratified solar atmosphere, including the photosphere. As the photons travel, they undergo scattering, absorption and re-emission. The opacity describes the attenuation of intensity of radiation through an optical medium, in this case the solar atmosphere, at a given wavelength. The probability of a photon escaping the solar atmosphere is very high when the optical depth at that wavelength equals unity. The solar photosphere is defined as the surface at which the optical depth of photons at a wavelength of 5,000 Å reaches unity.

The temperature of gas continues to drop with height until a temperature-minimum of about 4,000 K is reached at 500 km above the surface. The decrease of temperature with height causes the noticeable effect that the limb of the solar disk appears darker than the disk-center in visible light, known as limb-darkening. Radiation from the solar disk center corresponds to the deeper and hotter layers of the photosphere, from whence most photons escape. Toward the limb, however, the radiation is predominantly from the upper layers of the photosphere, which are cooler.

Overall, the radiation from the Sun is close to a blackbody spectrum from the near ultraviolet to infrared wavelength. Superimposed on this spectrum are numerous absorption lines from different atomic species in the solar atmosphere. The absorption lines carry a wealth of information about the medium in which they are formed. The strength and shape of a spectral line is influenced by temperature, electron pressure, gas pressure, magnetic field, abundance, and velocity. A proper understanding and interpretation of these lines provides important information on the properties of the solar atmosphere.

The absorption lines were first observed as thin dark bands against the colorful solar spectrum by William H. Wollaston, a chemist and physicist from England, in 1802. These lines were later rediscovered by Joseph Fraunhofer in 1814. He labeled the strongest spectral lines alphabetically, a practice that is still in use. By correlating the laboratory spectra with the Fraunhofer lines in the solar spectra, Kirchoff and Bunsen (Kirchhoff & Bunsen, 1860) discovered the presence of chemical elements in the Sun. Moore et al. (1966) gave one of the first detailed identifications of solar spectral lines from 3,000 Å to 9,000 Å.

The chemical composition of the Sun forms a basis for the determination of elemental abundances of almost all objects in the universe, from small planets to large galaxies. Since the beginning of the 20th century, the spectral lines have been used to determine the abundances of different elements on the Sun, starting with the pioneering work of Russell (1929) followed by Goldberg et al. (1960). Asplund et al. (2009) provide a comprehensive and homogeneous compilation of present-day photospheric abundances of all the elements up to atomic number 92. Hydrogen (12) is the most abundant element, followed by He (10.93±0.01), while traces of C (8.43±0.05), N (7.83±0.05), O (8.69±0.05), Ne (7.93±0.1), Mg (7.60±0.04), Si (7.51±0.03), S (7.12±0.03), and Fe (7.50±0.04) are common among other elements.2 To infer such photospheric abundances from the observed solar spectrum, realistic models of the solar atmosphere and details of the formation of spectral lines are crucial. In addition, precise atomic and molecular data are necessary (see also review by Grevesse et al., 2007).

Radiation reveals various characteristics of the gas. By observing the splitting of spectral lines and circular polarization of the components due to the Zeeman effect, Hale (1908a, b) discovered the presence of a magnetic field on the Sun. For the deduction of atmospheric quantities such as temperature, velocity stratification, and magnetic field vector, a detailed modeling of the spectral lines is necessary. This involves solving the radiative transfer equation discussed in sect. 2.1.

The propagation of radiation through the solar atmosphere is given by the radiative transfer equation. For a plane-parallel atmosphere, it can be written as

$Display mathematics$
(1)

where $μ=cos(θ)$, with $θ$ being the heliocentric angle, $I(λ,τ,μ)$ is the specific intensity, $S(λ,τ,μ)$ is the source function, and $τ$ is the optical depth. If $k(λ,τ)$ is the absorption coefficient, then $dτ=kdz$, where $dz$ is the geometrical height in the atmosphere. The source function represents the radiation from the surface of the Sun, which gets absorbed or scattered by the cooler material in the atmosphere. In a purely absorbing medium, the atomic-level populations are given by the Saha–Boltzmann statistics at the local temperature. The absorption line is said to be formed in local thermodynamic equilibrium, or LTE. In this case, the source function in Equation 1 is the same as the Planck function $(Bλ)$. In a scattering medium, the assumption of LTE fails and the spectral-line profiles are formed in non-LTE (NLTE) conditions. The atomic-level populations can no longer be determined using the Saha–Boltzmann statistics but are to be obtained by solving the statistical-equilibrium equation. The source function in the NLTE case is given by

$Display mathematics$
(2)

where $Jλ$ is the mean intensity and $ϵ$ is the photon destruction probability defined as

$Display mathematics$
(3)

$Cul$ and $Aul$ are the Einstein coefficients for transition between upper-level $u$ and lower-level $l$. When the density of the medium is large, then $Cul$ is much greater than $Aul$ such that $ϵ$ is close to unity. The source function is then equal to the Planck function and the medium is in LTE. However, in most of the astrophysical plasma, including the solar atmosphere, $ϵ<1$. Both scattering and absorption should be taken into account for a proper modeling of the spectral line profiles. For solar photospheric lines, the assumption of LTE is a good approximation, as the plasma density is large. The rest of this section concentrates mainly on the LTE treatment of spectral lines. For more details on the radiative transfer and NLTE effects, see Chandrasekhar (1950), Athay (1972), Rybicki and Lightman (1986), Rutten (2003), and Hubeny and Mihalas (2014).

The scattering of photons in the solar atmosphere and the magnetic field polarizes the solar radiation. While the magnetic field induces both linear and circular polarization, scattering produces only linear polarization. The polarization state of radiation is defined by the four Stokes parameters. They were first introduced by Sir George Stokes in 1852 and later brought into extensive usage by Chandrasekhar (1950) (see also textbooks on polarization by Stenflo, 1994; Landi Degl’Innocenti, 2014). Polarized radiation is represented as

$Display mathematics$
(4)

$I$ is known as the Stokes vector, with $I$ being the intensity, and $Q,U$, and $V$ measuring the linear polarization, inclination of plane of polarization, and circular polarization, respectively. One of the main advantages of such a representation is its ability to define partially polarized light.

The anisotropy of the solar radiation induces scattering polarization in spectral lines. Due to limb-darkening, the anisotropy is maximum at the solar limb and decreases as one moves toward the disk center due to symmetry. This is purely an NLTE effect. For lines formed in LTE, such as the photospheric lines, the medium is collision dominated and scattering polarization is rare. However, numerical computations suggest that lines such as the Sr i 4607 Å absorption line, formed in the upper photosphere with a large scattering polarization signal at the limb, has a measurable $Q$ signal at the disk center due to horizontal inhomogenities within granules and intergranules produced by the solar convection (Trujillo Bueno, & Shchukina, 2007; del Pino Alemán et al., 2018). Attempts are now being made to measure these small polarization signals using state-of-the-art spectropolarimeters like the Fast Solar Polarimeter (FSP, Iglesias et al., 2016; Zeuner et al., 2018) and the Zürich IMaging POLarimeter (ZIMPOL, Ramelli et al., 2014; Bianda et al., 2018). In the presence of a magnetic field, the linear polarization induced by scattering is modified and the plane of polarization is rotated. This is known as the Hanle effect (Stenflo, 1982). Once again this effect is commonly observed in lines formed in the upper atmosphere like the chromosphere where the NLTE effects are dominant. In the photospheric lines, the main cause for linear as well as circular polarization is the Zeeman effect. This effect results in splitting of the spectral lines into multiple components depending on the strength and orientation of the magnetic field. On the Sun, it was first observed in a sunspot, leading to the discovery of the first magnetic field outside our planet (see also Section 3.2).

The radiative transfer equation for the Zeeman effect can be written as

$Display mathematics$
(5)

Here $I$ is the Stokes vector given in Equation 4, $S$ is the source vector. $E$ is a $4×4$ unity matrix, and $K$ is the absorption matrix defined as

$Display mathematics$
(6)

The quantities $ηI,Q,U,V$ and $ρQ,U,V$ are related to the properties of the propagating medium. These expressions were first derived using a classical theory by Unno (1956), Rachkovsky (1962), Jefferies et al. (1989), and later with a quantum mechanics approach by Landi Degl’Innocenti and Landi Degl’Innocenti (1972). Examples of the Stokes profiles in the quiet Sun and in a sunspot observed by the telescope Hinode at the two iron lines 6,301.5 Å and 6,302.5 Å are shown in Figure 2. Of the two lines, the 6,302.5 Å line is magnetically more sensitive than the 6,301.5 Å line, which is expressed in its larger Landé g-factor. In that sunspot, the splitting of the 6,302.5Å line can be seen in intensity. Also, this line has larger $Q,U$, and $V$ signals in the sunspot compared to the 6,301.5 Å line. In this figure, all the Stokes profiles are normalized to spatially averaged “quiet Sun” intensity in the continuum. For a detailed review on the effects of magnetic field on the spectral lines, see Solanki (1993).

Figure 2. Spectropolarimetric diagnostics of the solar photosphere near a sunspot. Shown in the top left is a continuum image of the sunspot near 6,300 Å. The dark umbra and the surrounding penunbra of the sunspot along with the solar granulation can be seen. The top right panels are the Stokes profiles covering the photospheric iron lines at 6,301.5 Å and 6,302.5 Å along the slit indicated by the black vertical line in the top left panel. Bottom panels show the line profiles at two locations along the slit (red: from the sunspot umbra; black: from a granule; both marked in the top right panels). In the bottom panels, $Ic$ is the spatially averaged “quiet Sun” intensity in the continuum. These observations were obtained with the space-based Hinode telescope on August 31, 2011.

## 2.3 Observations and Modeling

Observations of solar spectral lines and their identification dates back to the 19th century. With advances in instrumentation, these recordings became more systematic and the spectral lines were documented in atlases, with the Utrecht Atlas of Minnaert et al. (1940) being one of the first solar-spectrum atlases (see Doerr et al., 2016, for a detailed comparison between different solar atlases). A majority of the photospheric lines are from the iron atom. Some of the commonly observed photospheric lines include Fe i 5,250 Å, Fe i 5,247 Å, Fe i 6,301.5 Å, Fe i 6,302.5 Å, Fe i 6,173 Å, and the Fe i 1.56 μ‎m lines in the infrared $(1m=104Å)$.

### 2.3.1 Measurement of Atmospheric Quantities

The extraction of physical quantities from the spectral lines requires solving the radiative-transfer equation discussed in the previous sections. Alternatively, magnetic-field information can also be extracted directly from the observed Stokes profiles without detailed modeling, in particular for the photospheric lines. Techniques for doing this include

• Weak field approximation: When the Zeeman splitting due to the magnetic field is smaller than the Doppler width of the spectral line, the Taylor expansion of Stokes $V$ can be expressed in terms of the unsplit intensity:

$Display mathematics$
(7)

where $ΔλH$ is the Zeeman splitting given by

$Display mathematics$
(8)

with $geff$ being the effective Landé g-factor and $B$ the magnetic-field strength. Similar expressions can also be derived for Stokes $Q$ and $U$, which are proportional to the second derivative of the unsplit intensity. Equation 7 has found wide applications in measurement of the line-of-sight component of the magnetic field. It was first applied to the chromospheric Na i D$1$ and D$2$ lines at 5,896 Å and 5,890 Å by Stenflo et al. (1984). It was then applied to the photospheric Fe i lines by Solanki and Stenflo (1984) to study the properties of solar magnetic flux tubes.

• Center-of-gravity method: In this method, the line-of-sight component of the magnetic-field strength is derived by computing the difference between the centers of gravity of the left- and right-circularly polarized profiles. Proposed and developed by Semel (1967, 1970) and Rees and Semel (1979), this method can be summarized in the following equation:

$Display mathematics$
(9)

where $(I±V)COG$ are the centroids of the left- and right-circularly polarized components and $m$ and $e$ are the electron mass and charge, respectively. Uitenbroek (2003) demonstrated that it is independent of spectral resolution and is suitable for applications that require high throughput (e.g., high-resolution observations with short integration times).

• Magnetic line-ratio method: The magnetic line-ratio method (MLR), first proposed by Stenflo (1973), measures the intrinsic magnetic-field strength from the ratio of Stokes $V$ of two spectral lines. The two lines should form at the same height in the atmosphere and have the same temperature but different magnetic sensitivities and not be contaminated by neighboring spectral lines. The most widely used line pair for MLR is the Fe i 5,247 Å–5,250 Å, formed in the photosphere. Stenflo (1973) used this line pair to discover the presence of the kilogauss magnetic field in the solar network. This method has subsequently found wide applications and is used also on the photospheric infrared line pair at 1.56 μ‎m (see Solanki et al., 1992), although they are not as well-suited for MLR as the former pair due to the difference in their heights of formation. Recently, Smitha and Solanki (2017) identified two new line pairs, one in the visible wavelength range and the other in the infrared. These new pairs are formed in the photosphere and can measure the intrinsic magnetic-field strength better than the original pair at 5,247 Å–5,250 Å. The main advantage of this method is its ability to measure the intrinsic field strength independently of the spatial resolution of the observing instrument. This method works best for intermediate field strengths where the amplitude of Stokes $V$ is proportional to the magnetic-field strength. For reviews on MLR and its application, see Solanki (1993), Nordlund et al. (2009), Stenflo (2013), and Smitha and Solanki (2017).

### 2.3.2 Stokes Profile Modeling and Inversions

As discussed previously, inference of physical quantities such as temperature, magnetic field, and velocity requires solving the radiative-transfer equation. In this process, an initial simple realistic model atmosphere of the Sun is assumed, which serves as input for solving the transfer equation. The output Stokes profiles are compared with the observations and any differences between the two are corrected for by suitable modification of the input model atmosphere. This is repeated until an acceptable match (defined by various criteria) to the observed Stokes profiles is obtained. A simplified overview of the Stokes inversion procedure is outlined in the block diagram in Figure 3.

Figure 3. Block diagram giving an overview of the different steps involved in Stokes profile inversions.

The response functions (RFs in Figure 3) (Beckers & Milkey, 1975) measure the response of the line profiles to perturbations in atmospheric conditions such as temperature, magnetic field, and velocity. In every iteration of the inversion procedure, the RFs guide the tweaking of model atmosphere until a convergence is reached between the modeled and observed Stokes profiles. The Stokes Inversion based on Response functions, or popularly known as the SIR code, is one of the first inversion codes based on the RFs developed by Ruiz Cobo and del Toro Iniesta (1992). Following SIR, inversion codes such as SPINOR (Frutiger & Solanki, 1998; Frutiger et al., 2000), NICOLE (Socas-Navarro et al., 2000), and recently SNAPI (Milić & van Noort, 2018) and STiC (de la Cruz Rodríguez et al., 2019) have been developed that can handle depth dependence of model atmospheres. Though SIR and SPINOR can only invert line profiles formed in LTE, NICOLE, SNAPI, and STiC can invert NLTE profiles as well. Apart from these, a number of inversion codes based on a Milne–Eddington atmosphere have been developed. For a detailed review, see del Toro Iniesta and Ruiz Cobo (2016).

# 3. Magnetoconvection

The solar photosphere represents a layer where the interplay between convection and magnetic field creates a plethora of dynamic structures, all coupled to the radiation. These magnetoconvective processes manifest themselves as granulation and magnetic concentrations with a range of spatial scales. Interpretation of these multi-scale photospheric structures is made possible by the numerical simulations of magnetoconvection that self-consistently model the interaction between convection and magnetic fields from small to large spatial scales. Here some important features of magnetoconvection are presented from observations and numerical simulations.

## 3.1 Convection

The outer 200,000 km shell beneath the solar surface makes up the convection zone. Here the energy is transferred through the motion of the plasma. The condition for convective stability is given by the Schwarzschild criterion. In a stably stratified fluid medium, when a fluid element is adiabatically displaced in the radial direction, it returns to its initial position due to gravity. This results in an oscillatory motion of the fluid element, with gravity acting as the restoring force. These oscillations are known as internal gravity waves, or g modes. The solar convection zone, however, is unstable such that the fluid element continues to rise. Convection is an instability that arises in a stratified fluid where the system is heated from below. This is usually the case in stars where the stellar interiors are hotter than their surfaces. The result is a negative temperature gradient from the core to the surface. Under this condition, consider a blob of gas in the interior of a star which is initially in mechanical and thermal equilibrium with the ambient medium. If this blob is displaced by a small distance outward (i.e., toward the surface against the direction of gravity), it is hotter than the surrounding. As long as the blob doesn’t reach a state of equilibrium with its surroundings, it continues to be displaced outward due to buoyancy. Mechanical equilibrium will be reached in the sound crossing time, $τs$, whereas thermal equilibrium will be reached in the conduction timescale,$τc$. In the solar interior,$τs=τc$. Therefore, the blob remains in mechanical equilibrium with its surroundings while it is displaced adiabatically with almost no energy exchange with the ambient medium. Thus the blob continues to rise until it loses its thermal energy or exchanges its gas with its surroundings. This convective transport of energy overturns the hot and cold material in the aptly named convection zone of the Sun (see Stix, 2002, for advanced discussion on stellar convection). More details on the internal dynamics of the Sun are provided in Gizon and Käpylä (2019). Here the observational characteristics of convection near the solar surface are briefly discussed.

Figure 4. A full disk magnetogram of the Sun displaying the line of sight component of the magnetic field obtained with Helioseismic Magnetic Imager onboard the Solar Dynamics Observatory (Schou et al., 2012; Scherrer et al., 2012; Pesnell et al., 2012). This map was recorded on July 16, 2011. The dark- and light-shaded regions are the south (negative) and north (positive) polarities of the field, respectively. The extended patches of concentrated positive and negative magnetic polarities are the active regions that host sunspots and plages. As the active region evolves and decays, its field is more diffused and scattered. Far away from the active regions are quiet regions and magnetic networks, structured by the convection near the surface. The line of sight component of the magnetic field is saturated at ±300 G.

A prominent feature of photospheric convection is the granulation which is formed as a result of the overshooting of convective motions into the convectively stable solar photosphere. The granules are bright elements or blobs of plasma surrounded by darker intergranular lanes that are interconnected (see Figure 1). The center of a granule consists of hot rising plasma and the intergranular lanes are regions where the cooler plasma sinks beneath the surface. The average lifetime of granules is about 5–10 minutes and their typical size is ~1,000 km.

The visibility or the contrast of granulation is directly related to the temperature fluctuations due to convective energy transport. The contrast measurements are rather difficult as they basically depend on the stray light, introduced either in the instrument or in the Earth’s atmosphere. Raw measurements from space-based Hinode observations show that the granulation has a contrast of about 7% near the red part of the visible spectrum, which can increase to 14% after removing various effects (Danilovic et al., 2008). In the near-ultraviolet wavelengths at 2,000 Å, the granular contrast is as high as 33% (Hirzberger et al., 2010).

The up- and downward granular motions near the surface Doppler shift the emergent radiation. This is seen as the blueshifts and redshifts of the core of any photospheric absorption line profile. These flow speeds are of the order of 1 km s-1 in an averaged sense, although locally much higher speeds are present (Hirzberger, 2002) (cf. Figure 5). Due to a large Reynolds number (defined as the ratio of inertial forces to the viscous forces) of the order of $1012$ in the solar photosphere, the flows are highly turbulent. A turbulence spectrum of the photospheric flows was already observationally determined in the early 1950s (Richardson & Schwarzschild, 1950). The turbulence manifests itself as observable vortex rolls and tubes in the intergranular downdrafts in the intensity and Doppler images of the photosphere that last for about 5 minutes (e.g., Bonet et al., 2008; Steiner et al., 2010).

The convection displays velocity fluctuations also on larger spatial scales compared to the prominent granulation. This velocity fluctuation manifests itself as a cellular pattern that is ubiquitously distributed over the Sun on scales of about 30,000 km, termed supergranulation (Leighton et al., 1962). Supergranules have a lifetime of about 1 day. The intensity contrast from their center to boundary is very weak. For this reason, they are feeble in the intensity images, so that their clearest observable signature is that of the horizontal flow field. Unlike granules, they have weaker up and downflows of about 50 m s-1 to 100 m s-1. Their horizontal velocity fluctuations are larger and are in the range of 300 m s-1 to 500 m s-1. Supergranules play an important role in advecting the magnetic field from the cell interior to the boundary, creating a magnetic network (see Rincon & Rieutord, 2018, for an exhaustive review on supergranules).

## 3.2 Magnetism

Until the year 1908, Earth was the only object on which magnetic fields were known to be present. In his seminal work, Hale (1908a) discovered that sunspots, dark blemishes on the photosphere, harbor magnetic fields. While scanning through sunspots with a spectroheliograph, Hale observed that the spectral lines from the dark umbral region of a sunspot were split into components of doublets and triplets. He also discovered these components to be circularly polarized, which indicated the presence of the Zeeman effect in sunspots (demonstrated in Figure 2). By measuring the separation of the spectral components of iron lines, Hale calculated that the spot had a magnetic field strength of 2.9 kG (for comparison, Earth’s magnetic field strength is less than 1 G). The detection of a magnetic field on the Sun was an important milestone in the field of astronomy and astrophysics.

Solar magnetism has important consequences for Earth and the heliosphere in general. Solar magnetic field displays cyclic behavior, which is also shown by spots and other manifestations of solar activity (van Driel-Gesztelyi & Owens, 2019). A dynamo action is necessary to maintain the solar magnetic field against its Ohmic decay. The various aspects of dynamo are covered by Cameron (2019).

At the surface, sunspots have a spatial extent or diameter of about 10,000 km and some of the largest sunspots can extend up to 50,000 km. Sunspots are embedded in active regions with strong magnetic-field strengths in the photosphere ranging from 2 kG to 3 kG. The strong magnetic field in the sunspot inhibits the convective motions below the observable layers and quenches the transport of heat to the surface. The reduced energy flux through the spot results in the darkening of sunspots compared to their surroundings (cf. Figure 1). Sunspots typically last for several days to several weeks. The large-scale organization and concentration of the magnetic field that forms sunspots is disturbed by the action of convection. As a consequence of interaction with the relentless convection, the sunspots become less organized over several days and their field gradually decays (see Solanki, 2003; Rempel, 2019, for extended reviews on sunspots). Active regions also host magnetic pores, which are coherent structures of a strong magnetic field that appear on granular scales of 1,000 km. These are miniature versions of sunspots without a distinct penumbral structure. Similar to a sunspot, a pore appears dark due to the suppressed convective motions below its surface. Magnetic pores are also seen in the quiet Sun away from active regions. Besides sunspots and pores, plages, observed as a dense collection of small magnetic flux elements concentrated in intergranular lanes, are an important type of structure in and around active regions that appear with or without sunspots. Plages extend over tens of arcsec on the solar surface. Individual flux elements in plages possess magnetic-field strengths of 1.2–1.7 kG (Rabin, 1992; Rüedi et al., 1992). These strong fields lead to evacuation of gas in the magnetic concentrations. As a result, the optical surface in plage elements forms deeper compared to the optical surface in the surrounding granules. The lateral inflow of radiation through the hot walls of magnetic concentrations, known as the hot wall effect, make the plages appear brighter at the solar surface in spite of the suppressed convection inside the magnetic elements (see Spruit, 1976).

Away from the active regions that host sunspots, the magnetic field is more scattered and its dynamics are governed by the evolution of convection near the surface. The plasma that upwells in the center of a supergranule is transported toward its outer edges, where it returns to the deeper layers below the surface in concentrated downflow plumes. During this evolutionary phase, the plasma advects the magnetic field from the center of that supergranule to the outer downflow lanes, where it forms magnetic networks. In a similar way, within each supergranule, the granular flow advects the magnetic field and concentrates it in the downdrafts of intergranular lanes, forming the magnetic internetwork. The magnetic concentrations or elements are advected with granular flow speeds of the order of 1 km s-1 or with supergranular flow speeds of 0.3 km s-1 for network elements (e.g., Muller et al., 1994; Berger & Title, 1996). Similar to plages, network and internetwork magnetic elements appear brighter due to the hot wall effect.

Figure 5. Spectropolarimetric diagnostics of the solar photosphere at high spatial resolution at the center of the solar disk observed with the 1-meter balloon-borne Sunrise telescope on June 9, 2009. The left panel is a magnetogram displaying the highly structured magnetic field in the quiet Sun at a spatial resolution better than 100 km. In the quiet Sun, magnetic field is found as small-scale concentrations in the lanes between granules (compare with Figure 1). The magnetogram is saturated at ±100 G. The right panel shows the map of line of sight (Doppler) velocity saturated at ±4 km s-1. The blue shaded regions represent upwelling plasma in convective granules. The red shaded regions, mostly confined to narrow lanes, are the regions of plasma downflows in the intergranular lanes. The area covered by these maps is about 30,000 km×30,000 km, which is similar to that of a large sunspot or a supergranular cell.

The magnetic field at the surface can be derived from various techniques (discussed in “2. Radiation”). Modern space-based telescopes are able to routinely obtain the magnetic field not only of sunspots, but of the full visible solar disk at a spatial resolution of $∼1′′$ and cadence of 45 s to 720 s. A map of the line of sight component of the magnetic field of the visible solar disk is shown in Figure 4. This map, which is called a magnetogram, shows the distribution of north (positive) and south (negative) polarities of the magnetic field of various features observed at the solar surface, including the sunspots and plage in active regions, decaying active regions, magnetic network, and the quiet Sun.3

Until the early 1970s, it was widely believed that strong magnetic fields are confined only to active regions hosting sunspots. In his seminal work, Stenflo (1973) used a pair of Fe I lines near 5,250 Å and employed the magnetic line-ratio technique to discover kilogauss fields in the magnetic network. Although Stenflo’s line-ratio method pointed to the presence of kilogauss magnetic-field strengths in the network regions, the exact determination of the properties of the quiet Sun intergranular magnetic field has remained a major challenge until recently. This is mainly because the features could not be resolved and therefore additional assumptions such as the filling factor or a multicomponent atmosphere had to be invoked to retrieve the strength of small-scale magnetic concentrations from spectropolarimetric observations. The situation changed with the unprecedented observations by the 1-meter balloon-borne Sunrise telescope (Solanki et al., 2010; Barthol et al., 2011; Martínez Pillet et al., 2011; Berkefeld et al., 2011; Gandorfer et al., 2011), which captured the quiet Sun at a spatial resolution better than 100 km (see Figure 5). Lagg et al. (2010) used these sunrise observations and measured field strengths in excess of 1.5 kG in the intergranular magnetic concentrations. Recent advances made using high-resolution magnetograms are highlighted in Solanki et al. (2010) and Solanki et al. (2017). However, even at this resolution, not all internetwork elements are resolved, and likely not all of them have kG fields (Riethmüller et al., 2014).

In the photosphere, the equipartition field strength, assuming a balance between plasma and magnetic pressures, is only about 500 G. This is much lower than the strengths of 1.5 kG observed in the network and in strong-field internetwork flux concentrations. Parker (1978) suggested that the magnetic concentrations of a few hundred Gauss in the downdrafts suppress net heat transport and are subjected to strong cooling. This leads to an enhancement in the downdraft within the magnetic concentration and (partial) evacuation of plasma from it. The evacuated magnetic concentrations are compressed by the external gas pressure in a process called convective collapse, which results in the strong observed magnetic field strengths of 1.5 kG (see also Spruit, 1979). Fischer et al. (2009) provided the observational statistics of convective collapse events in the solar photosphere.

Furthermore, there is evidence for substantial hidden magnetic fields at spatial scales below the resolution of current observations. Trujillo Bueno et al. (2004) used modeling of scattering polarization and the Hanle effect and estimated an average magnetic field strength of 130 G in the quiet Sun. According to the current estimates, at least three orders of magnitude more magnetic flux resides in the quiet Sun than in the active regions (Stein, 2012).

The origin of magnetic field in the magnetic features, ranging from large sunspots to small concentrations in the intergranular lanes, is in the convection zone. For a magnetic structure embedded in the convective zone to be in pressure equilibrium with the surrounding gas, the sum of its internal gas pressure and magnetic pressure must be equal to the external gas pressure. This pressure equilibrium implies reduced gas density within the magnetic structure. As a result, the magnetic structure in the convection zone is subjected to buoyancy forces and ascends to the surface. The subsurface properties of the magnetic field, however, are not known due to a lack of direct observations. At the surface, however, one does observe that the magnetic flux patches emerge as simple or complex bipoles with a pair of opposite magnetic polarities. The presence of alternating positive and negative polarities can be seen in Figure 4. During the formation of active regions, flux emergence can last for several hours to days. In the case of smaller structures, the emergence timescales are much shorter (down to the granular lifetimes of 5–10 minutes). A detailed review of the theory of flux emergence is presented in Cheung and Isobe (2014).

The magnetic field at the surface forms a myriad of closed and open structures that extend beyond the photosphere (i.e., into the atmosphere of the Sun). The closed structures are those in which the magnetic field loops back to the surface, whereas in the open structures the field extends into the heliosphere. These magnetic fields and their evolution play a key role in (a) heating the solar atmosphere (Klimchuk, 2019), (b) driving the Sun’s activity, and (c) accelerating the solar wind and shaping the heliosphere (Cranmer, 2019; Owens, 2019; Arridge, 2019). When one moves away from a source of heat, the temperature gradually decreases, following the second law of thermodynamics. This is also the case in the Sun from its core to just above the surface, where the temperature reaches a minimum of about 4,500 K, consistent with thermodynamic laws. However, strangely enough, from observations it is found that beyond this layer of minimum temperature the plasma is heated to about 10,000 K, forming the solar chromosphere (Jafarzadeh, 2019). Beyond this, the temperature suddenly increases to a million Kelvin, resulting in the solar corona from where the extreme ultraviolet and X-ray emission from highly ionized atomic species is observed. This rise in temperature from the photosphere to the corona is one of the long-standing mysteries in the whole of astrophysics. It is known that the energy required to heat the plasma to such high temperatures in the solar atmosphere is delivered by the magnetic field. The turbulent surface convective motions stress the magnetic field, leading to a variety of magnetohydrodynamic (MHD) processes such as waves and current sheets that heat the solar atmosphere. In some extreme cases, excessive stressing of the field leads to a long-term (days) buildup and storage of large amounts of magnetic energy in the solar atmosphere. Under favorable conditions, this energy is explosively released in a short time (minutes to hours), triggering solar flares (Longcope, 2019) and coronal mass ejections (Green, 2019). These extreme events have the potential to cause geomagnetic storms, aurorae, and can even damage power grids as well as satellites orbiting Earth. Such events are more frequent during the phase when the solar surface is covered with more active regions, which often display complex morphology and evolution. Therefore, continued observations of photospheric magnetic fields is pivotal in broadening knowledge of the Sun–Earth connection.

## 3.3 Numerical Simulations

Numerical models are a key to decoding the physics of the solar photosphere and any astrophysical system, in general. Models of the photosphere combine the equations of hydrodynamics and electromagnetism in a unified framework of MHD to simulate the evolution of the plasma and magnetic fields (Priest, 2019b). State-of-the-art MHD models of the magnetoconvection include stratification, a realistic equation of state (a relation between the density and temperature of the plasma), partial ionization, and radiative transfer. The scope of such models is to simulate the evolution of the solar photosphere to reproduce the realistic interaction between convection and magnetic fields that give rise to radiative signatures, which can then be compared with the observations.

Figure 6. The solar photosphere reproduced with state-of-the-art numerical simulations. Top panel is a synthetic image of a simulated sunspot (including umbra, penumbra, and the surrounding quiet-Sun granulation) in the continuum near 6,300 Å. The field of view is 30,000 km×10,000 km. The details of these simulations are presented in Rempel (2012). Bottom panels are synthetic images of plage in the continuum near 5,250 Å (left) and 1.56 $μ$m (right). The field of view is 12,000 km×12,000 km and the average field strength is 200 G. All the snapshots are produced from a 3D radiation MHD code, MURaM. The SPINOR code was used for the synthesis of these continuum images.

The basic numerical methods to simulate solar convection and granulation have been laid out for several decades (e.g., Nordlund, 1982). Nordlund (1984) presented some of the first numerical simulations of the solar granulation. Solanki (1987) developed semi-empirical model atmospheres of magnetic flux tubes that explain the observed polarization signals in the photosphere (see Solanki, 1993, for a review on magnetic flux tube models). High-resolution three-dimensional (3D) hydrodynamic simulations applicable to solar convection (Cattaneo et al., 1991), including the effects of radiation, that reproduced observed granular properties and dynamics (e.g., velocities, temperature fluctuations) soon followed (Stein & Nordlund, 1998). Presently, there are a number of radiation MHD codes that can reliably simulate the convective properties of the solar photosphere (see Beeck et al., 2012, for a comparison between simulations produced from different codes).

Here, some of the results obtained with the MURaM code developed by Vögler et al. (2005) are discussed. The MURaM code has been widely used to model the magnetoconvective processes in the photosphere and the underlying layers. These models cover a variety of magnetic-field features, from the quiet Sun to smaller pores to larger sunspots (e.g., Schüssler & Vögler, 2006; Vögler & Schüssler, 2007; Cameron et al., 2007; Rempel & Cheung, 2014). Dynamic events such as the emergence of magnetic flux (e.g., Cheung et al., 2007; Tortosa-Andreu & Moreno-Insertis, 2009), MHD vortices, shocks, and waves (e.g., Moll et al., 2012; Shelyag et al., 2013) have also been studied extensively with MURaM.

The numerical simulations of the magnetoconvection have been quite successful over the years. The current models are able to provide both qualitative and quantitative comparisons with modern high-resolution observations. In Figure 6, synthetic continuum snapshots are displayed from MURaM simulations of sunspots and plage in the visible and near-infrared wavelength ranges. The qualitative similarities in the appearance of the umbra, penumbra, and the granulation in these models and the observations (cf. Figure 1) are remarkable (see Rempel, 2019, for a review on MHD models of sunspots). Danilovic et al. (2008) and Hirzberger et al. (2010) concluded that the intensity contrast of solar granulation from the MHD simulations is consistent with the observations, a crucial test validating the reliability of those models. These simulations are also able to reproduce the properties of photospheric magnetic fields such as their magnetic-energy power spectra, which are in excellent agreement with the observations (e.g., Rempel, 2014; Danilovic et al., 2016).

Realistic modeling of the convection near surface layers is important for an accurate determination of elemental abundances in the solar photosphere. Spectral lines synthesized from model atmospheres are compared with the observed solar spectrum to infer the chemical composition. A traditional approach is to use time-independent one-dimensional (1D) model atmospheres, assuming hydrostatic equilibrium to synthesize the spectral lines and continuum. The model developed by Holweger and Müller (1974) has been a very widely used 1D atmosphere in solar abundances studies. Such models generally assume line formation under local thermodynamic equilibrium (LTE). However, to properly account for the evolving solar convection, time-dependent 3D models of the surface layers are necessary, which should also include the effects of non-LTE in forward modeling the spectrum. Asplund et al. (2009) used 3D hydrodynamic models of the solar convection and approximation of non-LTE effects, which is arbitrary in most cases, to estimate photospheric chemical composition. Their calculations from 3D atmospheres lead to significantly lower abundances of C, N, O, and Ne compared to widely used values in earlier studies.

Overall, the importance of numerical simulations cannot be overstated. Models of magnetoconvection are far ahead of the observations in terms of the scales of the structures they resolve. At 2 km resolution (Rempel, 2014), the models are able to probe the photosphere with super-magnification compared to the observable structures at 100 km resolution. To this end, the models are well poised to be tested against high-resolution observations with next-generation telescopes. In particular, the height variation of plasma and magnetic-field properties from the simulations can be compared with the many-line observations of the photosphere. The numerical simulations together with the observations deepen the understanding of the solar photosphere.

# 4. Prospects

The studies of the solar photosphere have evolved over several centuries, from systematic observations of large sunspots to spatially resolved measurements of magnetic structures in the intergranular lanes that extend only 100 km. The future of photospheric studies will follow multiple avenues, from ultra-high-resolution observations, via many-line studies (which allow a better sampling of height), to large-scale and high-resolution radiation MHD simulations. Currently, ground-based and balloon-borne telescopes can reach a spatial resolution of roughly 100 km. Thanks to the upcoming solar facilities in the near future, such as the 4-meter Daniel K. Inouye Solar Telescope (DKIST; Elmore et al., 2014) and the planned 4-meter European Solar Telescope (Collados et al., 2013), the photosphere will be able to be resolved down to spatial resolutions of 30 km. This will further enable deeper probes into the turbulent magnetoconvective processes at unprecedented detail. In addition, ESA’s and NASA’s Solar Orbiter mission, along with ground-based telescopes, for the first time will provide stereoscopic imagery of the solar photosphere. Such a look at the photosphere is expected to reveal novel characteristics of magnetoconvection and of magnetohydrodynamic (MHD) waves. Until now, photospheric observations have been made from the vantage point of the Sun–Earth ecliptic plane. This means that the understanding of the photosphere is generally limited to the latitudes between $±70°$ of the Sun. Solar Orbiter will image the Sun from out of the ecliptic plane. These pioneering photospheric observations will reveal the properties of granulation, supergranulation, and the long-term evolution of the magnetic field at the solar poles.4 All these new details will add to our current understanding of solar dynamo, magnetic cycles, and solar activity.

Going forward into the realm of high-spatial and -temporal resolution observations, there are several features of photospheric dynamics to be scrutinized further. These concern mainly convection and magnetic fields. Here a list of a few questions is given that will likely be addressed with the upcoming solar facilities.

• • How does the turbulent interaction of plasma and magnetic fields proceed at small spatial scales?

• • What is the substructure of magnetic concentrations in ARs and in the quiet Sun?

• • How does the velocity field within these magnetic concentrations evolve?

• • What are the fine-scale features of the flux emergence from large sunspots to small pores?

• • How does the penumbra of a sunspot form and how does a sunspot decay?

• • How do the latitudinal properties of the quiet Sun magnetic field vary over a solar cycle?

• • How does the polar small-scale magnetic field vary with time?

This article has only provided a bird’s-eye view of various aspects of the solar photosphere. Interested readers can find more detailed information on these topics from many excellent textbooks, reviews, and research articles listed in the references.

# Acknowledgments

The authors thank the reviewers for useful comments that improved the presentation of this article. L.P.C. received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 707837. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 695075). This work was partly supported by the BK21 plus program through the National Research Foundation (NRF), funded by the Ministry of Education of Korea. Hinode is a Japanese mission developed and launched by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA, and STFC (U.K.) as international partners. Scientific operation of the Hinode mission is conducted by the Hinode science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partner countries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (U.K.), NASA (U.S.), ESA, and NSC (Norway). SDO data are courtesy of NASA/SDO and the AIA, and HMI science teams. The German contribution to Sunrise and its reflight was funded by the Max Planck Foundation, the Strategic Innovations Fund of the President of the Max Planck Society (MPG), DLR, and private donations by supporting members of the Max Planck Society, which is gratefully acknowledged. The Spanish contribution was funded by the Ministerio de Economía y Competitividad under Projects ESP2013-47349-C6 and ESP2014-56169-C6, partially using European FEDER funds. The HAO contribution was partly funded through NASA grant number NNX13AE95G. This project has made use of NASA’s Astrophysics Data System.

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## Notes:

(1.) For a historical timeline of solar physics, see online.

(2.) The numbers in parentheses are photospheric logarithmic abundances relative to hydrogen, where the logarithmic abundance for H is defined to be 12.

(3.) In solar physics, north and south polarities of the magnetic field are usually just called positive and negative (field lines directed out of the surface and into the surface).

(4.) NASA’s STEREO observatory, launched in 2006 and composed of a pair of spacecrafts, one orbiting ahead of Earth and the other trailing behind, has already delivered stereoscopic images of the solar corona.