Space Physics Summary, Sintesi del corso di Fisica Sperimentale

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Space Physics

Questions and Answers

Linda and Jin

Prof: Consolati

M.Sc Space Engineering

Academic Year 2024-

Contents

List of Figures

Introduction

This report addresses the optimal staging of a three-stage launch vehicle for a mission to a 300 km circular orbit. The required mission velocity is first determined, including the appropriate loss models, to define the design conditions. A multi-objective optimization is then performed to identify optimal mass distributions on a Pareto front, minimizing the initial mass while maximizing the payload mass. Finally, a sensitivity analysis is conducted to evaluate the influence of key structural and propulsion parameters on overall vehicle performance.

1 Astronomy

1.1 How we get information on the surface temperature of a star?

To determine the surface temperature of a star, we analyze its spectrum, which is the distribution of flux as a function of wavelength or frequency. The continuous spectrum determines the color of the star, and the color is directly related to its temperature. In first approximation, a star behaves like a blackbody, meaning it emits radiation that depends only on its temperature. Planck’s law describes the spectral energy distribution of a blackbody and shows how the emitted intensity depends on both wavelength and temperature. From Planck’s law we obtain two important results: the Stefan–Boltzmann law, which states that the total emitted energy increases as T 4 , and Wien’s displacement law,

λmax = b T

which tells us that the wavelength of maximum emission is inversely proportional to the temperature. Therefore, by observing the stellar spectrum and measuring the wavelength at which the emission peaks, we can estimate the surface temperature of the star. Although stars are not perfect blackbodies, their continuous spectrum is well approximated by blackbody radiation, allowing us to determine their temperature from spectral observations.

1.2 How can we measure the distance of a star?

To measure the distance of a star, the used method is stellar parallax. Parallax is the apparent shift in the position of a nearby star with respect to very distant background stars as the Earth moves around the Sun. The idea is purely geometric. We observe the star at two opposite points in the Earth’s orbit, separated by six months. The baseline of this measurement is the Earth–Sun distance, which is 1 astronomical unit (AU). Because of this change in viewpoint, the nearby star appears to shift slightly relative to very distant stars, which can be considered fixed. The parallax angle p is defined as half of the total angular shift measured over six months. For small angles, which is always the case in astronomy, we can use the approximation

tan p ≈ p ≈

1 AU

d

From this simple relation, we obtain

d =

1 AU

p

This shows that the distance is inversely proportional to the parallax angle: the smaller the angle, the larger the distance.

1.3 The magnitude of a star

The magnitude of a star is a logarithmic measure of its brightness. There are two important definitions: the apparent magnitude m, which is the brightness we measure from Earth, and the absolute magnitude M , which is defined as the magnitude a star would have if it were placed at a standard distance of 10 parsecs. The magnitude scale is logarithmic because the human eye see brightness in a logarithmic way. By comparing the flux of a star at distance d with the flux it would have at 10 parsecs, we obtain the distance modulus formula:

M = m + 5 − 5 log d,

where d is expressed in parsecs. This relation shows that if we measure the apparent magnitude and know either the distance or the absolute magnitude, we can determine the other quantity. Therefore, magnitude provides a direct link between observed brightness and intrinsic luminosity, once the distance is known.

1.7 How is it possible to get information on the mass of a star?

The mass of a star can usually not be measured directly. The most reliable method is to study binary systems, since about half of the stars in our galaxy are in binaries. In a binary system, two stars orbit around their common center of mass, and their motion allows us to determine their masses using Newton’s laws. If both stars are visually resolved, we can measure their orbital period and the size of the orbit, and then apply Kepler’s third law to obtain the total mass of the system. In spectroscopic binaries, the stars cannot be resolved, but we observe periodic shifts of their spectral lines due to the Doppler effect. From these velocity variations and the orbital period, we can determine the mass function and estimate the masses. In eclipsing binaries, the light curve shows periodic dips when one star passes in front of the other. Combining photometric and spectroscopic data gives very precise masses. In some cases, the spectrum shows two overlapping spectral types, indicating the presence of two stars. For single main-sequence stars, when no binary companion is available, we can use the empirical mass–luminosity relation, which shows that more massive stars are significantly more luminous. Therefore, binary systems pro- vide the most direct and accurate method to measure stellar masses, while empirical relations are used when binaries are not available.

1.8 Why in the core of the stars only fusion, and not fission processes occur?

Nuclear fusion and nuclear fission are two different nuclear processes that release energy. Fusion is the process in which two light nuclei combine to form a heavier nucleus. Fission, on the other hand, is the process in which a heavy nucleus splits into two lighter nuclei. In both cases, energy is released because the final nuclei are more stable and have a higher binding energy per nucleon than the initial ones. If we look at the binding energy curve, we see that it increases with mass number up to iron, which is the most stable element. This means that light nuclei, such as hydrogen and helium, can release energy by fusing together, moving toward more stable configurations. In contrast, fission releases energy only for very heavy nuclei, such as uranium, because splitting them produces nuclei closer to the stability peak. Stars are mainly composed of light elements, especially hydrogen. Therefore, fusion is energetically favorable in their cores, while fission is not. Moreover, the extreme temperature and pressure in stellar cores allow protons to overcome their electrostatic repulsion and approach closely enough for the strong nuclear force to bind them together. This leads to fusion reactions such as the proton–proton chain or the CNO cycle. For these reasons, stars generate energy through fusion and not through fission.

1.9 How can two protons ‘melt’ together?

Two protons repel each other because they are both positively charged. This electrostatic repulsion creates what is called the Coulomb barrier, which prevents them from approaching each other. In order for fusion to occur, the protons must come extremely close, where the strong nuclear force becomes dominant and can bind them together. If we consider only classical physics, the temperature required to overcome the Coulomb barrier would be extremely high, much higher than the temperature in the core of the Sun. The average thermal energy in the Sun’s core is not sufficient, in classical terms, to allow protons to collide with enough energy to fuse. However, there are two important effects that make fusion possible. First, particles in the stellar core do not all have the same energy; they follow a statistical distribution. Second, and most importantly, quantum tunneling allows protons to penetrate the Coulomb barrier even if they do not have enough classical energy to overcome it. In quantum mechanics, particles have wave-like properties, and there is a finite probability that they can cross an energy barrier. Therefore, fusion in stars occurs because of the combination of high temperature, the high number of particles, and quantum tunneling. Without tunneling, stellar fusion at solar temperatures would not be possible.

1.10 Explain the PPI chain

The proton–proton chain is the main energy production mechanism in stars like the Sun. It describes how hydrogen is transformed into helium in the stellar core and releases energy through nuclear fusion. The first step is: proton + proton gives deuterium, a positron, and a neutrino. In this step, one proton changes into a neutron through the weak interaction. This is the slowest reaction, and it controls how fast the star produces energy.

The second step is: deuterium + proton gives helium-3 and a gamma photon. This reaction is much faster. The third step is: helium-3 + helium-3 gives helium-4 and two protons. The two protons are released and can participate again in the chain. If we look at the global result, four protons are converted into one helium nucleus. A small amount of mass disappears and is transformed into energy, which is what makes the star shine. Some energy is carried away by neutrinos, but most of it heats the stellar core.

1.11 the PPII and PPIII chains

The PP-II and PP-III chains are alternative branches of the proton-proton process. They begin with the same first two reactions as PPI: first, proton + proton produces deuterium, a positron, and a neutrino; second, deuterium plus proton produces helium-3 and a gamma photon. After helium-3 is produced, the chain can follow different paths depending mainly on the temperature. In the PP-II branch, helium-3 + helium-4 to produce beryllium-7 and a gamma photon. Then beryllium-7 captures an electron and becomes lithium-7, emitting a neutrino. After that, lithium-7 reacts with a proton and forms an excited beryllium-8 while emitting a gamma photon. Finally, this excited beryllium-8 is unstable and quickly breaks into two helium-4 nuclei. So, in PP-II, the key intermediate nuclei are beryllium-7 and lithium-7, and an important point is that neutrinos are produced when beryllium-7 captures an electron. In the PP-III branch, the first steps are again the same: proton plus proton gives deuterium with a positron and a neutrino, and deuterium plus proton gives helium-3 with a gamma photon. Then helium-3 plus helium-4 again produces beryllium-7 and a gamma photon. At this point the path changes: beryllium-7 captures a proton and becomes boron-8, emitting a gamma photon. Then boron-8 decays into an excited beryllium-8 nucleus, producing a positron and a neutrino. Finally, the excited beryllium-8 rapidly decays into two helium-4 nuclei. Therefore, PP-III is characterized by the production of boron-8 and by higher-energy neutrinos. In general, PP-II and especially PP-III become more important at higher core temperatures than the PP-I branch.

1.12 The CNO cycle and the production of carbon

The CNO cycle, which stands for Carbon Nitrogen Oxygen cycle, is another nuclear fusion process that converts hydrogen into helium. It becomes dominant in stars more massive and hotter than the Sun, because it requires higher core temperatures than the proton–proton chain. The cycle begins when a carbon-12 nucleus captures a proton and becomes nitrogen-13, emitting a gamma photon. Nitrogen-13 is unstable and decays into carbon-13, producing a positron and a neutrino. Then carbon- 13 captures another proton and becomes nitrogen-14, again emitting a gamma photon. Nitrogen-14 captures another proton and forms oxygen-15. Oxygen-15 then decays into nitrogen-15, emitting a positron and a neutrino. Finally, nitrogen-15 captures one more proton and splits into a helium-4 nucleus and carbon-12. At the end of the cycle, carbon-12 is regenerated. This is why carbon acts as a catalyst: it helps the reaction occur but is not consumed. If we look at the global result, four protons are converted into one helium nucleus, just like in the proton–proton chain. The difference is that in the CNO cycle the reactions are much more sensitive to temperature, so the energy production increases very rapidly with increasing temperature. This is why the CNO cycle dominates in massive stars.

1.13 Which processes make possible the formation of the elements in the core of a star?

Elements in the core of a star are formed through nuclear fusion reactions that occur at very high temperatures. First, hydrogen is converted into helium through the proton–proton chain or the CNO cycle. When hydrogen is exhausted, the core contracts and becomes hotter. Then helium nuclei can fuse together in the triple-alpha process to form carbon. If the temperature increases further, helium can combine with carbon to form oxygen, and then progressively heavier elements such as neon and magnesium. This process is called alpha capture, because helium nuclei are added step by step. In very massive stars, even heavier elements can be formed through additional burning stages, such as carbon burning and oxygen burning. These reactions continue until iron is produced. Fusion stops at iron because forming elements heavier than iron does not release energy.

1.18 Why are there limiting values to the mass of a star?

There are limiting values to the mass of a star because different physical pressures can support a star only within certain mass ranges. There is a lower mass limit because if a protostar is too light, it cannot reach a core temperature high enough to ignite hydrogen fusion. In that case, it becomes a brown dwarf instead of a true star. There is also an upper mass limit. For very massive stars, radiation pressure becomes extremely strong. The intense radiation pushes outward on the outer layers of the star. If the mass is too large, radiation pressure drives powerful stellar winds that remove material from the star, preventing it from growing indefinitely. Finally, for compact stars such as white dwarfs, there is a maximum mass that electron degeneracy pressure can support. If this limit, known as the Chandrasekhar limit, is exceeded, gravitational collapse continues. Therefore, both gravity and different forms of pressure determine lower and upper limits for stellar masses.

1.19 Why Massive Stars Remain for a Shorter Time in the Main Sequence with Respect

to Small Stars (Orders of Magnitude)

Massive stars remain for a shorter time in the main sequence because they consume their nuclear fuel much more rapidly than small stars. During the main sequence, stars produce energy by burning hydrogen into helium in their cores. Although massive stars contain more hydrogen fuel, their luminosity increases very strongly with mass. In fact, luminosity scales approximately as mass to the power three or even higher. This means that if a star is ten times more massive than the Sun, it can be thousands of times more luminous. Since luminosity represents the rate at which energy is produced, massive stars burn their fuel much faster. The lifetime of a star on the main sequence is roughly proportional to the available fuel divided by the luminosity. Because luminosity increases faster than mass, the lifetime decreases as mass increases. In terms of orders of magnitude, a star like the Sun remains in the main sequence for about ten billion years. A star with fifteen solar masses lives only about ten million years. In contrast, a low-mass star of about one quarter solar mass can remain on the main sequence for tens of billions of years, longer than the current age of the Universe. Therefore, massive stars are short-lived because they are extremely luminous and consume their hydrogen fuel very rapidly.

1.20 Evolution of Stars with Mass Less or Comparable to that of the Sun

Stars with mass comparable to or less than the Sun evolve in a relatively gentle way. During the main sequence, they burn hydrogen into helium in the core. When hydrogen in the core is exhausted, fusion stops in the center and the core contracts under gravity. As the core contracts, it heats up, while hydrogen burning continues in a shell around the core. The outer layers expand and cool, and the star becomes a red giant. At this stage, the luminosity increases and the radius becomes much larger. When the core temperature becomes high enough, helium fusion begins through the triple-alpha process, producing carbon and oxygen. In Sun-like stars, helium ignition may occur suddenly in what is called the helium flash. After helium is exhausted, the star cannot reach temperatures high enough to fuse heavier elements like carbon. The core, composed mainly of carbon and oxygen, contracts and becomes supported by electron degeneracy pressure. The outer layers are expelled, forming a planetary nebula. The final remnant is a white dwarf, which is a compact object supported by electron degeneracy pressure. It slowly cools over time without further nuclear reactions.

1.21 Evolution of Massive Stars

Massive stars, with masses greater than about eight solar masses, evolve much more dramatically. Like all stars, they begin by burning hydrogen in the core during the main sequence phase. Because of their high mass, they are very luminous and short-lived. After hydrogen is exhausted, the core contracts and the temperature increases enough to ignite helium fusion. Unlike low-mass stars, massive stars can reach even higher temperatures, allowing them to fuse heavier elements in successive stages.

They undergo helium burning, carbon burning, neon burning, oxygen burning, and silicon burning. Each stage produces heavier elements, and the star develops an onion-like structure, with different burning shells surrounding the core. Eventually, the core becomes composed mainly of iron. Fusion of iron does not release energy, so no further energy can support the star against gravity. The iron core collapses rapidly, triggering a core-collapse supernova explosion. After the explosion, the remaining core may become a neutron star or, if the mass is very large, a black hole.

1.22 Possible Final Stages of Stars

The final stage of a star depends mainly on its initial mass. For low-mass stars, the final stage is a white dwarf. After expelling their outer layers as a planetary nebula, the remaining carbon–oxygen core is supported by electron degeneracy pressure. It does not undergo further fusion and slowly cools over time. For intermediate and massive stars, the core-collapse process after iron formation leads to a supernova explosion. If the remaining core mass is below a certain limit, it becomes a neutron star, supported by neutron degeneracy pressure. If the core mass is even larger, not even neutron degeneracy pressure can stop the collapse, and the object becomes a black hole. Therefore, the possible final stages are white dwarfs, neutron stars, or black holes, depending on the initial mass of the star.

1.23 Type Ia and Type II Supernovae

Type Ia and Type II supernovae are two different kinds of stellar explosions. A Type Ia supernova happens in a binary system. A white dwarf star pulls matter from a companion star. As it gains mass, it approaches a critical limit called the Chandrasekhar limit. When this limit is reached, the white dwarf becomes unstable and a sudden thermonuclear explosion occurs. The entire star is destroyed. In the spectrum of a Type Ia supernova, we do not see hydrogen lines. These supernovae have almost the same maximum brightness, so they are used to measure distances in the Universe. A Type II supernova happens in a massive star, much heavier than the Sun. When the star finishes burning elements in its core, it eventually forms an iron core. Since iron cannot produce energy by fusion, the core collapses under gravity. This collapse produces a powerful explosion that ejects the outer layers of the star. In this case, hydrogen lines are present in the spectrum. After the explosion, the remaining core becomes either a neutron star or a black hole. So, in simple terms: Type Ia comes from a white dwarf in a binary system and is a thermonuclear explosion, while Type II comes from the collapse of a massive star.

1.24 Main Properties of a Neutron Star

A neutron star is formed almost exclusively by neutrons. A neutron star is the compact left after a massive star explodes as a supernova. It is supported by neutron degeneracy pressure, which is a quantum effect that prevents further gravitational collapse. Neutron stars are extremely small, with a typical radius of about 12 km, but they contain a mass comparable to that of the Sun. This makes them extraordinarily dense, with densities similar to those of atomic nuclei. They also possess extremely strong surface gravity, very rapid rotation (sometimes with millisecond periods), and extremely intense magnetic fields. Because of their small size, high density, fast rotation, and strong magnetic field, neutron stars are among the most extreme objects in the Universe.

1.25 the Existence of Neutron Stars

The existence of neutron stars was confirmed in 1967 with the discovery of pulsars. Pulsars are objects that emit very regular radio pulses. These pulses are produced by a rapidly rotating neutron star. The magnetic axis of the star is not aligned with its rotation axis. Because of this, radiation is emitted in narrow beams from the magnetic poles. As the star rotates, these beams sweep through space like a lighthouse. When one of the beams points toward the Earth, we detect a pulse.

amounts of gas and dust, and star formation occurs mainly in the spiral arms. Their masses and luminosities are comparable to large elliptical galaxies. Irregular galaxies do not have a well-defined shape. They are generally smaller, with lower masses, and often contain gas and young stars, which means they can show active star formation.

1.31 Structure of the Milky Way: Disk, Halo, and Bulge

The Milky Way is a spiral galaxy composed of three main structural components: the disk, the bulge, and the halo. The disk is the flattened region that contains most of the Galaxy’s stars, gas, and dust. It includes the spiral arms, where active star formation takes place. The Sun is located in the disk, about 8 kiloparsecs from the Galactic center. The disk itself is divided into a thin disk, which contains younger stars, gas, and ongoing star formation, and a thick disk, which contains older stars and less gas. At the center of the Galaxy lies the bulge, a dense and roughly spherical concentration of stars. The bulge is composed mainly of older stars and has little gas compared to the disk. It represents an early phase of star formation in the history of the Galaxy. Surrounding both the disk and the bulge is the halo. The halo is a large, roughly spherical region that contains old stars and globular clusters. It extends far beyond the visible disk. The halo also contains most of the Galaxy’s dark matter, which dominates the total mass of the Milky Way.

1.32 The Central Region of the Milky Way

The central region of the Milky Way is very complex and difficult to observe because it is heavily obscured by gas and dust. For this reason, it is mainly studied using infrared, radio, and X-ray observations. At the very center lies a compact radio source called Sagittarius A star. Observations of stars orbiting around this region show that there is a supermassive black hole with a mass of about four million solar masses. The motions of nearby stars provide strong evidence for this compact object. The central region also contains a dense concentration of gas and dust, including a structure known as the circumnuclear ring. Star formation has occurred in the past, and the region can show energetic activity due to matter falling toward the central black hole. Although the Galactic center appears relatively quiet today, it likely experienced more active phases in the past.

1.33 Dark Matter: Clues to Its Existence

The main evidence for dark matter comes from the rotation curves of galaxies. If only visible matter were present, the orbital velocity of stars should decrease with distance from the Galactic center. However, observa- tions show that the rotation curve remains approximately constant even far from the center. This implies that there is additional unseen mass distributed throughout the Galaxy. Another clue comes from comparing the total gravitational mass of the Milky Way with the mass of its luminous components, such as stars, gas, and dust. The observed gravitational effects require much more mass than what is visible. Dark matter appears to be distributed in a large, roughly spherical halo surrounding the Galaxy. It does not emit or absorb light, and it interacts mainly through gravity. Current candidates include hypothetical particles, but its true nature is still unknown.

1.34 Exoplanet Detection Methods: Gravitational Microlensing

Gravitational microlensing is based on general relativity. When a star passes in front of a more distant back- ground star, its gravitational field bends the light of the background star. This bending causes a temporary increase in brightness, called a microlensing event. If the foreground star has a planet, the planet produces a small additional distortion in the light curve. This creates a short deviation in the brightness variation, allowing the planet to be detected. Microlensing does not depend on the light from the planet itself. It can detect low-mass planets and even planets far from their host stars. However, microlensing events are rare and do not repeat, because they require a very precise alignment between observer, lens, and source.

1.35 Exoplanet Detection Methods: Transit Technique

The transit method detects exoplanets by observing the small decrease in a star’s brightness when a planet passes in front of it. During a transit, the planet blocks a fraction of the star’s light, producing a periodic dip in the light curve. From the depth of the brightness drop, we can estimate the planet’s radius. From the time between repeated transits, we determine the orbital period. Using Kepler’s laws, we can estimate the distance between the planet and the star. This method is particularly effective for detecting close-in planets. However, it requires the orbital plane to be aligned with our line of sight. Not all planetary systems are aligned in this way.

1.36 Advantages and Limitations of Exoplanet Detection Methods

Different detection methods have different strengths and weaknesses. The radial velocity method measures the wobble of a star caused by an orbiting planet. It is very precise and allows us to estimate the planet’s mass. However, it is more sensitive to massive planets close to the star. The transit method allows us to measure the planet’s radius and, when combined with radial velocity, its density. It is efficient for detecting many planets at once, but it requires a favorable geometric alignment and does not directly provide the planet’s mass. Gravitational microlensing is sensitive to low-mass planets and planets far from their stars. It can even detect free-floating planets. However, events are rare, short-lived, and do not repeat. In summary, no single method is complete. Combining multiple techniques provides the most reliable information about exoplanets.

1.37 Possible Biosignatures for Life on Exoplanets

A habitable planet is a solid-surface world that can maintain liquid water on its surface. This depends on temperature and pressure, which are controlled by the star’s luminosity, the planet’s distance from the star, and the composition of its atmosphere. The region where liquid water can exist is called the habitable zone. In our Solar System, Mercury has almost no atmosphere and cannot retain heat. Venus has a thick CO 2 atmosphere causing a strong greenhouse effect and extremely high temperatures. Earth lies in the habitable zone and maintains liquid water due to a balanced atmosphere. Mars has a thin atmosphere and low pressure, so liquid water is unstable. The giant planets — Jupiter and Saturn (gas giants), and Uranus and Neptune (ice giants) — are composed mainly of hydrogen, helium, and ices. They do not have solid surfaces suitable for life as we know it, but their atmospheres contain molecules such as methane and ammonia. A biosignature is a detectable sign of life. Remote biosignatures are divided into three types: atmospheric, surface, and temporal. Atmospheric biosignatures include gases such as O 2 , O 3 , CH 4 , or combinations of oxidized and reduced gases out of chemical equilibrium. Surface biosignatures include the “red edge” caused by vegetation. Temporal biosignatures refer to periodic variations over time, such as seasonal changes in atmospheric composition. However, abiotic processes can sometimes mimic biological signals, so interpretation must be done carefully.

1.38 Apparatus for the Detection of Gravitational Waves

Gravitational waves are detected using laser interferometers based on the Michelson configuration. A laser beam is split into two perpendicular arms several kilometers long. The beams are reflected by mirrors and recombined. When a gravitational wave passes, it slightly stretches one arm and compresses the other. This produces a tiny change in the interference pattern of the recombined light. The strain is defined as h = ∆L/L. To increase sensitivity, detectors use Fabry–Perot cavities, high-power lasers, ultra-high vacuum systems, seismic isolation, and extremely precise mirrors. Examples include LIGO in the USA, VIRGO in Italy, KAGRA in Japan, and the future space mission LISA.

2 the Sun

2.1 Structure and Main Properties of the Sun

The Sun is a main-sequence star about 4.6 billion years old, powered by nuclear fusion in its core through the proton-proton (pp) chain, where hydrogen is converted into helium. It produces energy at a rate of about 4 times 10 power26 W. About 99% of this energy is generated within the inner quarter of the solar radius. The internal structure of the Sun is divided into three main regions:

1. Core: This is the central region, extending to about 0.25 solar radii. The temperature is about 15 million K. Nuclear fusion occurs here, producing helium and neutrinos.
2. Radiative Zone: Extends from the core to about 0.7 solar radii. Energy is transported outward by radiation. Photons are repeatedly absorbed and re-emitted, taking thousands to millions of years to escape.
3. Convective Zone: Extends from 0.7 solar radii to the surface. Here the temperature drops enough that radiation becomes inefficient, and energy is transported by convection: hot plasma rises, cools, and sinks. The Sun shows differential rotation: it rotates in about 25 days at the equator and about 30 days at the poles. This rotation pattern extends through much of the convective zone. The Sun also emits neutrinos from its core, providing direct evidence of nuclear fusion. The study of solar oscillations allows us to investigate the internal structure and rotation of the Sun.

2.2 The Solar Atmosphere

The solar atmosphere consists of three main layers: the photosphere, chromosphere, and corona. Photosphere: This is the visible surface of the Sun, about 300–500 km thick, with a temperature at the inner boundary of the photosphere is 8000 K and at the outer boundary 4500 K. It shows granulation due to convection below the surface. Sunspots are observed here. Chromosphere: Located above the photosphere, about 500 km thick. The temperature increases from about 4500 K to 6000 K. Normally this layer is not visible, because its radiation is much weaker than that of the photosphere. However, during total solar eclipses, the chromosphere shines into view for a few seconds, when the Moon hides the photosphere completely. During eclipses the chromospheric spectrum, called flash spectrum, can be observed: among several lines stand out the lines of hydrogen, helium and certain metals. It is possible to observe the chromosphere also outside an eclipse by using a detector,filtering the light to a certain wavelength. In this case, we observe bright and dark features: bright features are called plages, and are indicative of areas with high magnetic activity and elevated temperatures, while dark features are called cells, which are cooler areas that lie between the brighter network of plages, that form a network-like pattern across the Sun’s surface. This network is created by the boundaries of super granules,large convective cells that transport energy from the interior of the Sun to the surface. To observe these features, astronomers use instruments that can filter out specific wavelengths of light. Corona: The outermost layer, extending millions of kilometers. It has extremely low density but very high temperature, about 1–2 million K due to electric currents induced by changing magnetic fields in the photosphere or the low cooling efficiency of low-density plasma. It is visible during total eclipses as a bright halo. The corona continuously expands outward forming the solar wind, which carries charged particles through the solar system.

2.3 Sunspots

Sunspots are dark regions on the photosphere caused by strong magnetic fields. They consist of a central dark region called the umbra and a surrounding lighter penumbra. The temperature in sunspots is about 1500 K lower than the surrounding surface, which makes them appear dark. The strong magnetic field inhibits convection, reducing the upward transport of heat. The number of sunspots varies with an average period of 11 years, known as the solar cycle. The latitudinal migration of sunspots during the cycle produces the butterfly diagram. The full magnetic cycle of the Sun lasts 22 years, because the magnetic polarity reverses every 11 years. the Sun also show several other types of surface activity: • Faculae and Plages are local bright regions in the photosphere and chromosphere. • Flares are among the most spectacular solar phenomena, which can even lead to the violent eruption of gas outwards. They appear as bright flashes, lasting from one second to just under an hour. • Coronal Mass Ejections are powerful magnetic explosions in the corona. In a few tens of

minutes, billions of tons of matter may be expelled from the Sun. Interaction with the Earth magnetic field produces also in these cases auroras and significant disturbance to the telecommunication systems

2.4 How is it possible to measure the magnetic field associated to the sunspot?

The magnetic field of sunspots is measured using the Zeeman effect, which is the splitting of spectral lines in the presence of a magnetic field. Without a magnetic field, atoms produce single spectral lines. When a magnetic field is present, the energy levels split, and the spectral line divides into multiple components. If the magnetic field is parallel to the line of sight, we observe two circularly polarized components (longi- tudinal Zeeman effect). If the magnetic field is perpendicular to the line of sight, we observe three components with linear polarization (transverse Zeeman effect). The amount of splitting is proportional to the strength of the magnetic field, allowing us to determine both its intensity and direction.

2.5 Solar Activity Beyond Sunspots

The Sun exhibits several types of magnetic activity besides sunspots. Prominences are large loops of plasma extending into the corona, guided by magnetic fields. Solar flares are sudden releases of magnetic energy due to magnetic reconnection. They emit radiation across the electromagnetic spectrum and can affect Earth’s communication systems. Coronal Mass Ejections (CMEs) are massive expulsions of plasma from the corona. When directed toward Earth, they can disturb the magnetosphere and produce auroras. The Sun also continuously emits the solar wind, a stream of charged particles that fills the solar system.

2.6 Inner Structure of the Planets

The internal structure of the planets depends mainly on their mass and chemical composition, and it reflects the processes of differentiation that occurred during their formation. When planets formed from accreting material in the protoplanetary disk, heavier elements tended to sink toward the center under gravity, while lighter materials remained in outer layers. This led to a stratified internal structure. The terrestrial planets — Mercury, Venus, Earth, and Mars — are composed mainly of heavy elements such as iron, nickel, silicon, and oxygen. They have a differentiated structure consisting of an iron–nickel core, a silicate mantle, and a crust. In Earth, for example, the outer core is liquid while the inner core is solid, and the motion of conductive fluid in the outer core generates the planetary magnetic field through dynamo action. The internal structure of terrestrial planets can be studied using seismic waves, since longitudinal waves propagate through both solids and liquids, while transverse waves propagate only through solids. By analyzing wave propagation, we can infer the presence of liquid layers and density variations. In contrast, giant planets like Jupiter and Saturn are mainly composed of hydrogen and helium. They likely contain a relatively small rocky or icy core surrounded by a very deep envelope of molecular hydrogen. In the inner regions, under extremely high pressure, hydrogen becomes metallic and electrically conductive, which explains their strong magnetic fields. Uranus and Neptune, the so-called ice giants, have a different composition. They contain a rocky core surrounded by a mantle rich in water, ammonia, and methane ices, and an outer hydrogen–helium atmosphere. Their internal pressure may not be sufficient to form large regions of metallic hydrogen, and their magnetic fields have more complex geometries. Therefore, planetary internal structure varies significantly across the Solar System and reflects both com- position and formation conditions.

2.7 The atmospheres of the planets.

With the exception of Mercury, all major planets possess an atmosphere. The characteristics of a planetary atmosphere depend on the planet’s mass, temperature, chemical composition, and distance from the Sun. Venus has a very dense atmosphere composed mainly of carbon dioxide. The CO 2 causes an extreme greenhouse effect: infrared radiation emitted from the surface is trapped, leading to surface temperatures of about 750 K and surface pressures around 90 atmospheres. This makes Venus the hottest planet in the Solar System, despite not being the closest to the Sun.

The IMF plays a crucial role in solar–terrestrial interactions. When the IMF has a southward component relative to Earth’s magnetic field, magnetic reconnection can occur at the magnetopause. This allows energy and particles from the solar wind to enter the magnetosphere, triggering geomagnetic activity. Thus, the IMF is fundamental in space weather phenomena and in coupling the Sun to planetary magneto- spheres.

2.11 Termination of the Solar Wind

The solar wind is a continuous flow of charged particles coming from the Sun. It travels outward in all directions and fills a large region of space called the heliosphere. As the solar wind moves farther away from the Sun, it eventually meets the interstellar medium, which is the gas and magnetic field that exist between stars in our galaxy. At first, the solar wind moves faster than the speed of sound in that plasma, so it is supersonic. At a distance of about 80–100 AU, the solar wind suddenly slows down. This region is called the termination shock. Here, the solar wind changes from supersonic to subsonic speed. After that comes the heliosheath, where the solar wind is slower, hotter, and more disturbed. Finally, at the heliopause, the pressure of the solar wind becomes equal to the pressure of the interstellar medium. Beyond the heliopause, the Sun no longer dominates — interstellar space begins. The Voyager spacecraft crossed these regions and confirmed this structure experimentally.

2.12 Why Are Sunspots Dark?

Sunspots look dark because they are cooler than the rest of the solar surface. The normal temperature of the photosphere is about 5800 K, while inside a sunspot the temperature is around 1500 K lower. Since brightness depends strongly on temperature, even this decrease makes the region appear dark compared to the surroundings. The reason for this lower temperature is the presence of a very strong magnetic field inside sunspots, up to about 0.4–0.5 Tesla. This strong magnetic field blocks convection. Normally, convection brings hot plasma from inside the Sun up to the surface, carrying energy. But in sunspots, the magnetic field stops this upward motion. So less energy reaches the surface there. As a result, the region cools down and emits less light, which is why it appears dark.

2.13 What is the mechanism explaining the stronger magnetic field in the sunspots with

respect to that on the poles?

The magnetic field in sunspots is stronger because it is concentrated in small areas. Inside the Sun, magnetic fields are created by the dynamo process, due to differential rotation and convection in the convection zone. These motions stretch and twist magnetic field lines. In some regions, magnetic flux tubes form. These are bundles of magnetic field lines that become concen- trated and rise toward the surface. When they reach the photosphere, they appear as sunspots. Because the magnetic field lines are packed into a small region, the field strength becomes very large locally. At the poles, the magnetic field is part of the Sun’s global magnetic field. It is spread over a much larger area and is therefore weaker. So the key idea is: sunspots have strong magnetic flux concentration, while polar fields are more diffuse.

2.14 Importance of Solar Neutrinos and Their Sources

Solar neutrinos are particles produced in nuclear reactions in the Sun’s core. They are mainly created in the proton–proton chain, which is the main fusion process in the Sun. A smaller part comes from the CNO cycle. Neutrinos are very special because they interact very weakly with matter. This means they escape directly from the solar core and reach Earth in about 8 minutes. In contrast, photons produced in the core take thousands to millions of years to reach the surface because they are constantly absorbed and re-emitted. So solar neutrinos give us direct information about what is happening inside the Sun right now. They confirm that nuclear fusion is really happening in the core. They are extremely important to test stellar structure models.

2.15 The problem of solar neutrinos and its solution.

When scientists first tried to measure solar neutrinos, they found only about one third of the number predicted by theory. This was called the solar neutrino problem. At first, people thought maybe the solar models were wrong. But later it was discovered that neutrinos can change type during their travel from the Sun to the Earth. This is called neutrino oscillation. There are three types of neutrinos: electron, muon, and tau. The Sun produces mainly electron neutrinos. However, during their journey, some electron neutrinos transform into muon or tau neutrinos. Early detectors could only detect electron neutrinos. So they missed the others. Later experiments that could detect all types of neutrinos showed that the total number matches theoretical predictions. This solved the problem and also proved that neutrinos have mass.

2.16 Methods to detect neutrinos.

Neutrinos are very difficult to detect because they interact very weakly with matter. So detectors must be very large and placed deep underground to avoid interference from cosmic rays. One method is the radiochemical method. In this technique, neutrinos hit specific atoms, like chlorine or gallium, and change them into different elements. Scientists then count how many atoms were transformed. Another method is the water Cherenkov detector. When a neutrino interacts in water, it produces a charged particle that moves faster than light in water. This creates Cherenkov radiation, which looks like a faint blue light and can be detected. There are also scintillation detectors, where special materials produce light when particles pass through them. These experiments allow scientists to measure neutrino flux and study neutrino oscillations.