Stars and The Sun VSIs

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We are all made of atoms created in stars!
Author

Stephen J. Mildenhall

Published

2024-04-12

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Notes from Stars and The Sun VSIs.

Stars

  • Spectral lines and gaps (caused by absorption by elements in the outer layers of the star)
  • p10: \(L\) luminosity - light energy emitted, \(\propto\) surface area \(\times T^4 \propto 4\pi R^2T^4\)
  • p18: \(PV=kNT\) means \(P=(kN/V)T\propto\) density \(\times T=\rho T\)
  • \(T_*\) central temperature and Eddington limit; consider a particle rotating at radius \(R\), temperature is \(\propto\) kinetic energy, and Newtonian mechanics says velocity in orbit is \(GM/R\), so \(T_*\propto |v|^2\propto GM/R\), giving inverse relationship between central temperature and radius
  • For sun we can estimate \(R\) (parallax) and \(M\) (orbits of planets) and get \(T_*=10^7\mathrm K\)
  • p24: Virial theorem \(2T+U=0\) so \(E=U+T=-T\); if the star radiates energy becomes more negative and \(T\) increases “a star heats up as it tries to cool” (radiate, contract, lower internal pressure, contracts further, gravitational potential energy converted into heat, increased fusion energy, heats up)
  • p27: thermal time scale = time to “cool down” = energy \(E\) / rate of radiation = about 30M years for sun
  • p37: fusion: 4H to He results in loss of 0.007 of the mass; sun generates \(3.8\times 10^{26}\)W, 1kg of H to He produces \(6\times 10^{14}\)J, so conversion rate is \(6.3\times 10^{11}\) kg/sec (vs mass of \(2\times 10^{30}\)kg) with a mass loss of only 0.007 of this amount
  • p41: Eddington limit
  • p42: (GPT explanation.) If the central temperature is too low to produce nuclear luminosity the star will shrink in an attempt to provide the luminosity from its binding energy. This refers to a star’s response to its internal conditions, particularly when the central temperature isn’t high enough to sustain nuclear fusio n at a rate that would balance the outward pressure and inward gravitational pull. Here’s what it means:
    1. Nuclear Luminosity: This term refers to the brightness of a star generated by nuclear fusion reactions in its core. For main sequence stars, this is primarily the fusion of hydrogen into helium.
    2. Central Temperature Too Low: If the core temperature of a star drops below the threshold needed for nuclear fusion, the energy production rate decreases. This reduction in energy output can’t counterbalance the gravitational forces trying to compress the star.
    3. Star Shrinks: In response to reduced pressure from within (due to lower energy production), the star contracts. This contraction increases the star’s central temperature and density because of the gravitational energy being converted into thermal energy.
    4. Luminosity from Binding Energy: As the star contracts, it converts gravitational potential energy into thermal energy, increasing the star’s core temperature. This process, known as the Kelvin-Helmholtz mechanism, was thought to be a significant energy source for the Sun before the discovery of nuclear fusion. The star effectively ‘shines’ by converting its gravitational binding energy into thermal energy and then light. This mechanism is particularly relevant for stars in their pre-main-sequence phase or for stars transitioning to later stages of their lifecycle, where nuclear fusion is not the primary energy source. It illustrates the star’s attempt to achieve hydrostatic equilibrium, balancing gravitational collapse with internal pressure.

GPT Notes

Main Sequence Stars

In stars, especially main sequence stars, there are fundamental relationships between temperature, mass, and radius. These relationships are crucial for understanding stellar evolution and characteristics. Here are the main relationships:

  1. Mass-Luminosity Relationship: For main sequence stars, there is a well-known relationship between mass and luminosity, which indirectly ties to temperature. More massive stars are more luminous and tend to have higher surface temperatures. The relationship is approximately \(L \propto M^{3.5}\) for stars on the main sequence, where \(L\) is luminosity and \(M\) is mass.
  2. Mass-Radius Relationship: Main sequence stars exhibit a mass-radius relationship, where more massive stars generally have larger radii. This relationship can vary depending on the star’s composition and age but is roughly \(R \propto M^{0.8}\) for many main sequence stars, where \(R\) is the radius.
  3. Temperature-Radius-Luminosity Relationship: The luminosity of a star is also related to its radius and surface temperature by the Stefan-Boltzmann law: \(L = 4\pi R^2 \sigma T^4\), where \(\sigma\) is the Stefan-Boltzmann constant, and \(T\) is the surface temperature. This implies that at a given temperature, larger stars are more luminous.
  4. Mass-Temperature Relationship: Generally, more massive main sequence stars have higher core temperatures, which facilitates higher rates of nuclear fusion, contributing to their greater luminosity and higher surface temperatures.

These relationships apply primarily to main sequence stars, which are in a stable phase of hydrogen burning. Stars in other phases of their lifecycle (like giants or white dwarfs) do not necessarily follow these relationships due to different internal processes and structures.

Virial Theorem

The virial theorem in astrophysics, particularly for stars, relates the kinetic energy (K) and potential energy (U) of a star system in a state of equilibrium. It states that, for a self-gravitating, stable star system:

\(2K + U = 0\)

Here, \(K\) is the total kinetic energy of the system, often associated with the thermal energy of the particles (like ions and electrons in stars), and \(U\) is the total potential gravitational energy.

This theorem is pivotal in understanding the energy balance and stability of stars. It implies that the internal thermal pressure (related to kinetic energy) balancing the gravitational collapse (related to potential energy) in stars maintains their stable structure. For a star in equilibrium, the virial theorem ensures that any disturbance in this balance would lead to adjustments, leading the star back to a stable equilibrium state.

Why factor 2

The factor of 2 in the virial theorem arises from the mathematical derivation of the theorem, specifically from the integration of the gravitational potential energy in a system of particles. When you calculate the gravitational potential energy (U) of a system, you sum over the potential energies of all pairs of particles, which involves an integral that ultimately introduces a factor of \(\frac{1}{2}\). However, when you calculate the kinetic energy (K), you sum over all individual particles without this pairwise factor.

In the context of the virial theorem, when you relate these two energies (kinetic and potential) for a system in equilibrium, the factor of 2 emerges from this difference in how each type of energy is calculated. The theorem is a result of applying the principles of conservation of energy and equilibrium dynamics to the system, leading to the relation \(2K + U = 0\). This relationship indicates that, for a stable, self-gravitating system, the kinetic energy (related to motion and thermal energy) is always half the magnitude (but positive) of the potential energy (related to the gravitational interactions), giving rise to the factor of 2.

The Eddington luminosity, or Eddington limit, is a fundamental concept in astrophysics that defines the maximum luminosity a star (or any astronomical object) can achieve when there is a balance between the outward radiation pressure and the inward gravitational pull acting on the ionized gas within the star. Here’s a detailed breakdown:

Eddington luminosity

  1. Definition: The Eddington luminosity is expressed as: \[ L_{\text{Edd}} = \frac{4\pi G M m_p c}{\sigma_T} \] where \(G\) is the gravitational constant, \(M\) is the mass of the star, \(m_p\) is the mass of the proton, \(c\) is the speed of light, and \(\sigma_T\) is the Thomson scattering cross-section for the electron.
  2. Implications: When a star’s luminosity exceeds the Eddington limit, the radiation pressure pushing outward overwhelms the gravitational force pulling inward. This can lead to powerful stellar winds or even the ejection of material from the star.
  3. Relevance: The Eddington limit is particularly significant in the study of massive stars, accreting black holes, and active galactic nuclei, where intense radiation fields are present.
  4. Temperature: The Eddington limit itself doesn’t directly specify a temperature; rather, it defines a luminosity threshold based on mass. However, objects approaching this limit are often extremely hot and emit vast amounts of radiation across a broad spectrum, including high-energy photons.

The Eddington limit is a critical threshold in astrophysics, beyond which the nature of radiation and matter interaction dramatically changes, influencing the evolution and behavior of stars and other luminous objects.

The Sun

Stars VSI makes it seem we have a good understanding of stars. The Sun VSI calls that into question, largely because we observe so much more about the sun and so see more that needs explaining. TL;DR: conducting plasmas are incredibly complicated!

  • p.10 Helium was discovered in spectral lines from the Sun
  • The sun gets hotter beyond its surface, in the corona
  • p.56 rate of rotation varies with latitude!
  • p.84 solar margeto-hydrodynamics and Alfvén’s theorem
  • p.88 magnetic lines wrap around (varying rotation) but must stay connected; eventually they explode out of the surface of the sun; magnetic polarity reverses with sun spot cycle
  • p.93-95: helioseismolgy and resonances of the sphere = same as electrons! (spherical harmonics)
  • p.104 solar X and γ-radiation creates ozone in upper atmosphere, which then protects us from the radiation (it is absorbed as the ozone is created)
  • p.128-9 solar flares and ejection of matter
  • Eugene Parker astronomer

GPT Notes

The Sun’s Atmosphere

The sun’s atmosphere is divided into several layers, primarily the photosphere, chromosphere, and corona, each exhibiting unique variations in temperature and pressure as one moves from the surface of the sun to outer space.

Photosphere

  • Temperature: Approximately 5,500 degrees Celsius
  • Pressure: Highest among the three layers, decreases with altitude
  • Description: The photosphere is the lowest layer of the sun’s atmosphere and the part we visually perceive as the sun’s surface. It is about 500 km thick. This layer is cooler and denser than the layers above.

Chromosphere

  • Temperature: Ranges from about 4,000 degrees Celsius at the bottom to around 25,000 degrees Celsius at the top
  • Pressure: Lower than the photosphere, continues decreasing with altitude
  • Description: The chromosphere is above the photosphere and is about 2,000 to 3,000 km thick. This layer is visible as a red rim during solar eclipses due to the hydrogen alpha spectral line it emits.

Corona

  • Temperature: Unusually high, ranging from 1 million to 3 million degrees Celsius
  • Pressure: Significantly lower than both the photosphere and chromosphere
  • Description: The corona is the outermost layer of the sun’s atmosphere, extending millions of kilometers into space. Despite its higher temperature, it appears faint due to its very low density.

Temperature and Pressure Variations: - Moving from the surface of the sun (photosphere) to outer space through the chromosphere and into the corona, there is a dramatic increase in temperature despite the decreasing density and pressure. This counterintuitive increase in the corona’s temperature, compared to the cooler underlying layers, is a major area of research in solar physics. The prevailing hypothesis involves the dissipation of magnetic energy into thermal energy through processes like magnetic reconnection and wave heating.

Coronium

The “coronium” mystery is a fascinating chapter in the history of solar physics, centered around the origin of mysterious spectral lines observed in the sun’s corona. This mystery was eventually resolved through the development of quantum mechanics and advancements in spectroscopy, leading to significant breakthroughs in understanding stellar atmospheres.

Background

During solar eclipses in the 19th century, astronomers observed spectral lines in the light from the sun’s corona that could not be matched with any known chemical elements. These lines were specifically prominent during the total solar eclipses when the sun’s disk is completely covered, allowing its faint corona to be visible. One of these lines, a bright green line, was particularly puzzling.

The Coronium Hypothesis

Initially, scientists hypothesized the existence of a new element called “coronium” to explain these mysterious spectral lines. It was thought to be an element lighter than hydrogen and only existing in the extreme conditions of the solar corona. This idea was akin to the earlier hypothesis of “helium,” which was first identified in the solar spectrum before it was found on Earth.

Resolution

The mystery of coronium persisted until the 1930s when advancements in atomic physics and quantum mechanics provided new insights into the behavior of highly ionized atoms (atoms that have lost many of their electrons). Walter Grotrian and Bengt Edlén were instrumental in resolving the coronium mystery. They suggested that the strange spectral lines were not due to new elements but rather highly ionized forms of well-known elements, which could exist only under the extreme temperatures and conditions found in the corona.

  • Walter Grotrian (1939): Grotrian mapped the observed green line to a transition in highly ionized iron (Fe XIII), indicating iron that has lost twelve of its electrons.
  • Bengt Edlén (1942): Using the new theoretical framework provided by quantum mechanics, Edlén identified several coronal lines, including the green line, as transitions in highly ionized states of iron, nickel, and calcium. These elements were ionized to such an extent that could only occur at the extremely high temperatures of the solar corona.

Conclusion

The resolution of the coronium mystery significantly advanced the field of spectroscopy and stellar physics, illustrating how elements under different physical conditions could exhibit vastly different spectral lines. It highlighted the importance of temperature and density in determining the ionization states of atoms in stellar atmospheres and paved the way for understanding stellar and solar phenomena more deeply. This revelation also reinforced the utility of quantum mechanics in explaining astrophysical phenomena and marked a major advancement in solar science.

Sun Spots and Weather

Sunspots themselves do not have a direct impact on Earth’s weather, but the solar activity associated with sunspots can influence the Earth’s space environment and, subtly, its climate. Here’s how it works:

Solar Activity and Sunspots

Sunspots are darker, cooler areas on the sun’s surface that have intense magnetic activity. They are often associated with solar phenomena such as solar flares and coronal mass ejections (CMEs). These features are indicators of the Sun’s magnetic activity, which varies in an approximately 11-year cycle, known as the solar cycle.

Impacts on Earth

  • Solar Radiation: During periods of high solar activity, which often correspond with a high number of sunspots, the sun emits increased ultraviolet (UV) and X-ray radiation. This increased radiation can cause the Earth’s upper atmosphere to heat up and expand.

  • Earth’s Upper Atmosphere: The heating and expansion of the upper atmosphere affect satellite orbits, GPS signals, and radio communications. This can also lead to increased drag on satellites in low Earth orbit, affecting their trajectory and lifespan.

  • Geomagnetic Storms: Solar flares and CMEs associated with sunspots can lead to geomagnetic storms when they interact with Earth’s magnetosphere. These storms can disrupt power grids, communication networks, and even cause auroras (northern and southern lights).

Subtle Climatic Effects

  • Atmospheric Chemistry: Changes in solar ultraviolet radiation can alter the chemical composition of the Earth’s stratosphere, particularly affecting ozone levels. This can influence atmospheric circulation patterns subtly.

  • Solar Irradiance: The slight changes in solar irradiance (the sun’s output of energy) during a solar cycle can potentially affect Earth’s climate. However, the effects are much subtler compared to other climatic drivers like greenhouse gases.

While sunspots and the solar activity associated with them can influence the Earth’s space weather significantly, their impact on Earth’s weather and climate is much more indirect and subtle, primarily affecting the upper atmosphere and only slightly influencing global climate patterns over longer timescales.

The Weber

The weber (symbol: Wb) is the SI unit of magnetic flux. A weber is defined as the magnetic flux that, linking a circuit of one turn, produces in it an electromotive force of one volt if it is uniformly reduced to zero within one second. In terms of base SI units, one weber is equivalent to one volt-second (V·s). More formally, it can be expressed in other base units as:

\[\text{Weber (Wb)} = \text{Volt (V)} \times \text{Second (s)} = \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2 \cdot \text{A}}\]

Here, kg is kilograms, m is meters, s is seconds, and A is amperes. This unit is crucial in the study of electromagnetism, particularly in contexts involving electromagnetic induction and magnetic circuits.

Plasmas

In a plasma, not all electrons are necessarily stripped from their atoms; rather, the degree of ionization can vary widely depending on the temperature and density of the plasma. Plasma is essentially a state of matter in which enough energy has been provided to free electrons from atoms or molecules, creating a mix of charged particles including ions and electrons. Here’s how it typically breaks down:

  1. Partially Ionized Plasma: This type of plasma contains a significant number of neutral atoms as well as ionized atoms and free electrons. The ionization level might be low, meaning that only a small fraction of the atoms have lost one or more electrons. This is common in cooler plasmas, such as those found in neon lights or certain types of astrophysical phenomena.

  2. Fully Ionized Plasma: In this case, nearly all atoms are ionized, meaning that almost all the electrons are stripped from their atoms. This occurs at very high temperatures, typically found in stars (including the sun), where the thermal energy is sufficient to overcome the electrons’ binding energy completely.

  3. Degree of Ionization: The term ‘degree of ionization’ refers to the proportion of atoms that have lost at least one electron, and it varies depending on the plasma’s temperature and density. Higher temperatures and lower densities generally increase the degree of ionization.

  4. Examples:

    • Astrophysical Plasma: In stars, the plasma is usually fully ionized due to the extremely high temperatures in the stellar core and corona.
    • Technological Plasma: In man-made devices like fluorescent lamps or plasma TVs, the plasma is typically only partially ionized.

Therefore, the extent to which electrons are stripped from atoms in a plasma can vary, reflecting the environmental conditions and energy levels present within the plasma.

Energy required to remove electrons

Stripping all electrons from atoms to create a fully ionized plasma for each element—hydrogen (H), helium (He), neon (Ne), and argon (Ar)—requires reaching or exceeding the ionization energy for each of their electrons. The amount of energy required to ionize an atom completely is specific to the element and its electronic structure. Here’s a simplified explanation using ionization energies, focusing on the energy needed to remove the last electron to achieve a fully ionized state:

1. Hydrogen (H)

  • Electrons: 1
  • Ionization Energy for Last Electron: 13.6 eV
  • Temperature Equivalent: Approximately 158,000 K

2. Helium (He)

  • Electrons: 2
  • Ionization Energy for Last Electron: 54.4 eV (from He+ to He2+)
  • Temperature Equivalent: Approximately 630,000 K

3. Neon (Ne)

  • Electrons: 10
  • Ionization Energy for Last Electron: Approximately 1262.2 eV (from Ne9+ to Ne10+)
  • Temperature Equivalent: About 14.6 million K

4. Argon (Ar)

  • Electrons: 18
  • Ionization Energy for Last Electron: Approximately 4226.2 eV (from Ar17+ to Ar18+)
  • Temperature Equivalent: About 49.1 million K

Calculating Temperature Equivalents

The temperature equivalents for each ionization energy can be estimated using the Boltzmann constant \((k \approx 8.617 \times 10^{-5} \text{ eV/K}\)). The temperature \(T\) required to provide sufficient thermal energy \(kT\) that compares with the ionization energy can be approximated by: \[ T \approx \frac{E}{k}. \]

Where \(E\) is the ionization energy for the last electron. I will now perform the calculations to give more precise values for these temperatures.

Here are the approximate temperatures required to strip all electrons from the atoms in a gas of hydrogen, helium, neon, and argon, resulting in fully ionized plasma:

  1. Hydrogen (H): Approximately 157,821 K
  2. Helium (He): Approximately 631,286 K
  3. Neon (Ne): Approximately 14,647,223 K
  4. Argon (Ar): Approximately 49,043,014 K

These temperatures highlight the extreme conditions needed to fully ionize these elements, particularly for heavier noble gases like neon and argon.

To ionize all atoms of hydrogen (H), helium (He), neon (Ne), and argon (Ar) by removing just one electron from each, we consider the ionization energy needed to remove the first electron from their neutral state. These ionization energies are lower than those required for complete ionization, as we’re only dealing with the first ionization process for each element:

First Ionization Energies

  • Hydrogen (H): 13.6 eV
  • Helium (He): 24.6 eV
  • Neon (Ne): 21.6 eV
  • Argon (Ar): 15.8 eV

Calculating Temperature Equivalents

The temperature equivalent for each first ionization energy can be estimated using the same approach as previously described, converting ionization energy (in electron volts) to a temperature (in Kelvin) using the Boltzmann constant. Let’s calculate those temperatures.

Here are the approximate temperatures required to ionize all atoms of hydrogen, helium, neon, and argon by removing just one electron from each:

  1. Hydrogen (H): Approximately 157,821 K
  2. Helium (He): Approximately 285,471 K
  3. Neon (Ne): Approximately 250,658 K
  4. Argon (Ar): Approximately 183,351 K

These temperatures indicate the minimum thermal energy required to initiate the ionization process for each element, by stripping away the first electron.

For Fe XIII

Fe XIII (iron ions with twelve electrons removed, resulting in a +12 charge) can be created on Earth, but it requires extremely high-energy environments to achieve such a high level of ionization. These conditions are typically found in laboratories using specialized equipment like:

  • Particle accelerators: These devices can accelerate ions to high energies and then smash them into targets or other particles, stripping away electrons in the process.
  • Fusion experiments: Research facilities focusing on nuclear fusion, like tokamaks or inertial confinement fusion devices, reach temperatures comparable to those in the sun’s corona, which can produce highly ionized plasma states including Fe XIII.
  • X-ray and ultraviolet light sources: High-intensity beams can provide enough energy to ionize iron up to the required state.

Creating and studying such highly ionized atoms helps in understanding astrophysical processes and improving techniques in spectroscopy and materials science.

To determine the total energy required to form an Fe^13+ ion, which is iron stripped of thirteen electrons, we need to consider the sum of the ionization energies for each step from the neutral Fe atom (Fe^0) to Fe^13+. This involves the energy required to remove each electron sequentially up to the 13th electron.

Ionization Energies for Iron (Fe): - The first few ionization energies of iron (from Fe^0 to Fe^1+, Fe^1+ to Fe^2+, and so on) increase as more electrons are removed, especially as one begins to remove electrons closer to the nucleus after the outermost electrons are gone.

Calculation Method: The total energy required to ionize neutral iron up to Fe^13+ is calculated by summing these ionization energies. I’ll provide these energies for the first thirteen electrons:

  1. 7.902 eV (Fe^0 to Fe^1+)
  2. 16.18 eV (Fe^1+ to Fe^2+)
  3. 30.651 eV (Fe^2+ to Fe^3+)
  4. 54.91 eV (Fe^3+ to Fe^4+)
  5. 75.0 eV (Fe^4+ to Fe^5+)
  6. 99.0 eV (Fe^5+ to Fe^6+)
  7. 125 eV (Fe^6+ to Fe^7+)
  8. 151 eV (Fe^7+ to Fe^8+)
  9. 233 eV (Fe^8+ to Fe^9+)
  10. 262 eV (Fe^9+ to Fe^10+)
  11. 290 eV (Fe^10+ to Fe^11+)
  12. 330 eV (Fe^11+ to Fe^12+)
  13. 350 eV (Fe^12+ to Fe^13+)

Now, let’s calculate the total ionization energy required to form Fe^13+.

The total energy required to form an Fe^13+ ion, which involves removing thirteen electrons from a neutral iron atom, is approximately 2024.643 electron volts (eV).

Hottest Temperatures in the Universe

The hottest temperatures observed in the current universe are typically found in high-energy astrophysical phenomena and advanced experimental physics facilities. Here are some contexts:

Natural Phenomena

  1. Core of Massive Stars: The cores of the most massive stars can reach temperatures exceeding 50 million Kelvin, particularly in the late stages of stellar evolution just before supernova events.

  2. Supernovae and Neutron Star Collisions: During these explosive events, temperatures can spike to billions of Kelvin, albeit briefly. For instance, when neutron stars collide, the temperatures can soar to several billion Kelvin, facilitating the synthesis of heavy elements.

  3. Accretion Disks Around Black Holes: Material in the accretion disks of black holes, particularly those at the centers of galaxies, can heat up to millions of Kelvin due to intense gravitational and frictional forces.

Experimental Settings

  1. Large Hadron Collider (LHC): In particle accelerator experiments, such as those conducted at the LHC, temperatures can reach over a trillion Kelvin for a very short duration when heavy ions (like lead ions) collide. These conditions mimic those just moments after the Big Bang, allowing physicists to study fundamental particles and forces.

  2. Fusion Experiments: In experiments aiming to achieve nuclear fusion, such as those conducted at the National Ignition Facility (NIF) or ITER (when it becomes fully operational), the core temperature of the plasma can exceed 100 million Kelvin, necessary for achieving conditions where fusion reactions can occur sustainably.

These temperatures dwarf those needed for full ionization of elements like hydrogen, helium, neon, and argon in laboratory plasmas, providing context for the extreme conditions under which different physical processes occur, both naturally in the universe and in controlled experimental settings.

Spherical Harmonics

The quantum state of electrons in an atom is determined by a set of quantum numbers, each describing a specific characteristic of the electron’s behavior and its arrangement around the nucleus. Here are the four primary quantum numbers that define the state of an electron in an atom:

  1. Principal Quantum Number (n): Indicates the main energy level or shell of the electron and determines the electron’s energy. It can take positive integer values (n = 1, 2, 3, …). The larger the value of n, the higher the energy level and the farther the electron is on average from the nucleus.

  2. Angular Momentum Quantum Number (l): Defines the shape of the orbital that electrons occupy and is dependent on the principal quantum number. It can take integer values from 0 to n-1 for each value of n. Each value of l corresponds to a different type of orbital (s, p, d, f, etc.), where l = 0 is an s-orbital, l = 1 is a p-orbital, and so on.

  3. Magnetic Quantum Number (m_l): Specifies the orientation of the orbital in space relative to the other orbitals and is influenced by the external magnetic field. It can take integer values from -l to +l, including zero.

  4. Spin Quantum Number (s): Describes the intrinsic spin of the electron. It can have only two values, +1/2 or -1/2, indicating the two possible orientations of the electron’s spin.

Deriving the Number of Electrons in Each Shell

To calculate the total number of electrons that can occupy each shell (energy level), you can use the quantum numbers as follows:

  • For each value of n, l ranges from 0 to n-1.
  • For each value of l, m_l ranges from -l to +l.
  • Each m_l value can be occupied by two electrons, one with spin +1/2 and the other with spin -1/2.

Therefore, the maximum number of electrons per shell can be calculated by summing the number of electrons that can occupy each subshell (defined by l), which is \(2(2l+1)\) electrons per subshell, summed over all possible l values for that n.

Let’s calculate the number of electrons in each shell for the first few values of n: - For n = 1 (1s shell): l = 0; m_l = 0; two electrons (one orbital × 2 spins) - For n = 2 (2s, 2p shells): - l = 0: m_l = 0; two electrons - l = 1: m_l = -1, 0, +1; six electrons (three orbitals × 2 spins) - For n = 3 (3s, 3p, 3d shells): - l = 0: m_l = 0; two electrons - l = 1: m_l = -1, 0, +1; six electrons - l = 2: m_l = -2, -1, 0, +1, +2; ten electrons (five orbitals × 2 spins)

Summing these, we find: - For n = 1: 2 electrons - For n = 2: 2 + 6 = 8 electrons - For n = 3: 2 + 6 + 10 = 18 electrons

This pattern continues as you increase n, with each shell able to hold more electrons due to the increasing number of subshells and the corresponding magnetic quantum numbers.

Ozone

Yes, ozone is produced by sunlight (solar radiation) in the upper atmosphere, particularly in the stratosphere. This production is a key component of the Earth’s ozone layer, which plays a critical role in shielding the planet from the Sun’s harmful ultraviolet (UV) radiation. Here’s how the process works:

Formation of Ozone

  1. Photodissociation of Oxygen Molecules: High-energy UV radiation from the Sun strikes oxygen molecules (O2) in the upper atmosphere. This radiation has enough energy to break the strong O-O bond in molecular oxygen, splitting it into two separate oxygen atoms (atomic oxygen, O). \[ O_2 + \text{UV light} \rightarrow 2O. \]

  2. Reaction with Molecular Oxygen: Each of these highly reactive atomic oxygens then quickly combines with another molecular oxygen under the presence of a third body (M), usually another oxygen molecule or nitrogen molecule, which carries away the excess energy without altering the final products. This reaction forms ozone (O3). \[ O + O_2 + M \rightarrow O_3 + M. \]

Importance of Ozone

  • UV Radiation Absorption: Ozone absorbs the majority of the Sun’s high-energy UV radiation, specifically UV-B and UV-C radiation. This absorption not only protects living organisms from UV-induced damage but also heats the stratosphere—an effect essential for atmospheric stability.
  • Dynamic Balance: The concentration of ozone in the stratosphere is the result of a balance between its production by the reaction mentioned above and its destruction through various catalytic reactions, often involving radicals such as chlorine (Cl) and bromine (Br), derived from man-made compounds.

The production of ozone by sunlight is thus a crucial natural process that forms a protective ozone layer approximately 10 to 50 kilometers above the Earth’s surface in the stratosphere.

Alfvén’s Theorem

Alfvén’s theorem, also known as the frozen-in flux theorem, is a fundamental principle in magnetohydrodynamics (MHD), named after Hannes Alfvén, a Swedish physicist who won the Nobel Prize in Physics in 1970 for his work in this field. The theorem describes the behavior of magnetic fields in highly conductive fluids like plasmas.

Key Aspects of Alfvén’s Theorem

  1. Magnetic Field Lines Move with the Fluid: Alfvén’s theorem states that in a perfectly conducting fluid, magnetic field lines are “frozen” into the fluid. This means that the magnetic field lines are carried along by the fluid flow due to the high electrical conductivity of the medium. In practical terms, if a fluid element moves, the magnetic field lines passing through this element move with it.

  2. Conservation of Magnetic Flux: The magnetic flux through any closed loop moving with the fluid remains constant over time. As the fluid moves and deforms, the magnetic field lines are deformed but not disconnected or broken. This conservation of magnetic flux is a critical concept in understanding phenomena such as the generation and evolution of magnetic fields in astrophysical objects like stars and galaxies.

  3. Implications for Plasma Physics: The theorem implies that any changes in the magnetic field within a highly conductive plasma are due to the motion of the plasma itself. This interrelation between plasma flows and magnetic fields is central to the dynamics of solar flares, the shaping of the solar corona, and the containment of plasma in fusion reactors.

Mathematical Formulation

Alfvén’s theorem can be mathematically expressed using the induction equation in magnetohydrodynamics, derived from Maxwell’s equations under the assumption of a perfectly conducting fluid: \[ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) \] where: - \(\mathbf{B}\) is the magnetic field, - \(\mathbf{v}\) is the velocity field of the fluid, - \(\nabla \times (\mathbf{v} \times \mathbf{B})\) represents the curl of the cross product of the velocity and magnetic fields.

This equation shows that the time evolution of the magnetic field \(\mathbf{B}\) is governed by its advection through the fluid flow \(\mathbf{v}\).

Applications and Significance

Alfvén’s theorem is pivotal in astrophysics, geophysics, and the design of magnetic confinement devices in fusion energy research. It helps explain the stability of magnetic fields in stars, the formation of jets and accretion disks around black holes, and is instrumental in studying Earth’s magnetosphere and solar wind interactions.

In summary, Alfvén’s theorem illustrates the deep connection between fluid dynamics and electromagnetic fields in conducting fluids, highlighting the complex interplay between plasma motions and magnetic fields in various cosmic and terrestrial environments.

Inviscid

The term “inviscid” refers to a fluid that has no viscosity. Viscosity is a measure of a fluid’s resistance to gradual deformation by shear stress or tensile stress. In practical terms, viscosity is what makes syrup slow-moving and water relatively quick to flow; it’s essentially the “thickness” or internal friction of a fluid.

Characteristics of Inviscid Fluids:

  • No Viscosity: Inviscid fluids are idealized fluids with zero viscosity, which means there is no internal friction between adjacent layers of the fluid as they move relative to each other.
  • Theoretical Concept: True inviscid fluids do not exist in reality; all real fluids have some level of viscosity. The concept of an inviscid fluid is used in fluid dynamics to simplify the equations of motion by neglecting the effects of viscosity.
  • Simplified Mathematical Modeling: Inviscid flow allows for the use of simpler fluid dynamics equations, such as the Euler equations, which do not include terms that account for viscous effects. This can be useful for analyzing situations where viscous forces are so small that they have negligible effects on the flow.

Applications and Examples:

  • Idealized Analysis: The concept of inviscid flow is often used in theoretical studies and preliminary design calculations where the influence of viscosity is considered negligible compared to other forces like inertia or pressure changes.
  • High-Speed Aerodynamics: In high-speed aerodynamics, especially at high altitudes where air viscosity can be considered relatively low compared to the inertial forces, the assumption of inviscid flow can be a reasonable approximation.
  • Potential Flow Theory: Inviscid flow is a central assumption in potential flow theory, which is used extensively in aerodynamics and hydrodynamics to predict flow patterns around objects, like airfoils and ship hulls, without considering viscous effects.

Limitations:

While the assumption of inviscid flow can simplify many problems in fluid dynamics, it is important to understand its limitations. For example, inviscid models cannot predict any behavior where viscosity plays a critical role, such as in the boundary layer close to solid surfaces or in predicting drag forces on objects.

In summary, while inviscid fluids are a theoretical concept, they are widely used in fluid dynamics to simplify problems and gain insights into the behavior of fluids under conditions where viscous effects are minimal or can be reasonably neglected.

Solar Dynamo

The “solar dynamo” refers to the process that generates the Sun’s magnetic field, driven by the complex movement of conducting plasma within the solar interior. This phenomenon is a key aspect of solar physics because it explains not only the existence of the Sun’s magnetic field but also its changes over time, including the approximately 11-year solar activity cycle known as the sunspot cycle.

Fundamental Concepts of the Solar Dynamo

  1. Plasma and Magnetic Fields: The Sun is composed primarily of plasma, a state of matter in which electrons are freed from their atomic nuclei, allowing the gas to conduct electricity. Moving electric charges generate magnetic fields, so the motion of the Sun’s plasma is inherently linked to magnetic phenomena.

  2. Differential Rotation: The Sun does not rotate uniformly; its equator rotates faster than its poles. This differential rotation stretches and twists magnetic field lines running through the solar plasma, amplifying and reorienting the magnetic fields.

  3. Convection and Turbulence: The solar plasma is not static but undergoes turbulent convection as heat moves outward from the Sun’s core. This convection occurs in a pattern of rising hot plasma and sinking cooler plasma. As plasma flows, it carries magnetic fields with it, leading to further complexity in the magnetic field structure.

  4. Poloidal and Toroidal Fields: The solar dynamo theory describes the transformation of the Sun’s magnetic field between two configurations:

    • Poloidal Field: A magnetic field that runs north-south along the Sun’s axis of rotation.
    • Toroidal Field: A magnetic field that loops east-west around the Sun’s equator, lying parallel to the solar surface.

The Dynamo Process

  • Generation of Toroidal Field: The differential rotation of the Sun shears the existing poloidal magnetic field, stretching and wrapping it around the equator to create a toroidal field.
  • Generation of Poloidal Field: Through mechanisms like the Babcock-Leighton process, the toroidal field is converted back into the poloidal field. In this process, the decay of sunspots and the rise of magnetic flux due to buoyancy contribute to the reformation of the poloidal field, influenced by the Coriolis force due to the Sun’s rotation.

Solar Activity Cycle

  • Sunspot Cycle: The solar dynamo is directly responsible for the solar activity cycle, most evident in the periodic rise and fall of sunspot numbers. Sunspots are regions of intense magnetic activity where the magnetic field lines are twisted and emerge through the solar surface, inhibiting convection and resulting in cooler, darker areas.
  • Polarity Reversal: Approximately every 11 years, the Sun’s magnetic field undergoes a polarity reversal, marking the peak of the solar cycle. This reversal is a key aspect of the dynamo process, where the poloidal field regenerates with opposite polarity, and a new cycle begins.

Importance of the Solar Dynamo Theory

Understanding the solar dynamo is crucial for predicting solar activity and its effects on space weather, which can impact satellite communications, power grids, and astronauts in space. The dynamo theory also provides a framework for studying similar magnetic phenomena in other stars and astrophysical objects, enhancing our understanding of stellar and galactic magnetism.

Parker Probe

The Parker Solar Probe is a groundbreaking mission launched by NASA on August 12, 2018, designed to study the Sun more closely than any spacecraft before it. Named after solar physicist Eugene Parker, who first theorized the existence of the solar wind in 1958, this mission aims to provide new insights into the behavior of the solar corona and the dynamics of solar wind, as well as to explore the structure and dynamics of the Sun’s magnetic fields.

Key Objectives:

  1. Trace the Flow of Energy and Heat: The probe aims to trace how energy and heat move through the solar corona and to explore what accelerates the solar wind as well as solar energetic particles.
  2. Study the Structure and Dynamics of Magnetic Fields: Understanding the structure and dynamics of the magnetic fields at the sources of solar wind.
  3. Explore Dusty Plasma Environment: Investigate the dusty plasma environment near the Sun and its influence on solar wind and energy propagation.

Technical and Design Innovations:

  • Heat Shield: One of the Parker Solar Probe’s most critical features is its heat shield, known as the Thermal Protection System. This shield enables the spacecraft to survive temperatures exceeding 1,377 degrees Celsius (2,500 degrees Fahrenheit) while keeping its instruments at a tolerable near-room temperature.
  • Close Solar Passes: The probe uses Venus’ gravity with multiple flybys to gradually bring its orbit closer to the Sun. It is set to come within 6.16 million kilometers (3.83 million miles) of the Sun’s surface, which is about seven times closer than any previous spacecraft.
  • High-Speed Journey: The Parker Solar Probe is one of the fastest man-made objects ever built, expected to reach speeds of up to 700,000 kilometers per hour (430,000 miles per hour) relative to the Sun at its closest approach.

Scientific Instruments:

The spacecraft carries a suite of scientific instruments designed to measure various aspects of the solar environment: - FIELDS: Measures electric and magnetic fields, radio emissions, and shock waves in the Sun’s atmospheric plasma. - WISPR (Wide Field Imager for Solar Probe): Takes images of the solar corona and solar wind. - SWEAP (Solar Wind Electrons Alphas and Protons Investigation): Collects data on the particles making up the solar wind. - ISʘIS (Integrated Science Investigation of the Sun): Measures particles across a wide range of energies.

Scientific Impact and Discoveries:

The Parker Solar Probe has already made significant discoveries since its launch: - Solar Wind Mystery: It provided new insights into the origin of the solar wind’s acceleration, which is not steady as previously thought but is highly dynamic and variable. - Switchbacks: Detected sudden, rogue waves within the solar wind that reverse direction like a zig-zag whipped back on itself, known as “switchbacks.” - Dust-Free Zone: It observed a decrease in cosmic dust near the Sun, hinting at a “dust-free” zone that had been hypothesized but not confirmed.

The ongoing mission is expected to continue providing groundbreaking data on solar processes that are fundamental to understanding both the Sun’s behavior and our broader universe. This mission represents a monumental leap forward in solar science, bringing humanity closer than ever to our star.

European Solar Orbiter

The European Solar Orbiter is a mission by the European Space Agency (ESA) in collaboration with NASA, launched on February 10, 2020. Its primary goal is to study the Sun up close, but from a unique perspective compared to other missions like NASA’s Parker Solar Probe. While the Parker Solar Probe dives close to the Sun to study its outer atmosphere directly, the Solar Orbiter combines close approach with a high inclination orbit around the Sun. This allows it to observe not just the equatorial regions like previous solar observatories but also the polar regions and the Sun’s far side, which are less studied areas.

Key Objectives:

  1. Solar and Heliospheric Science: The Solar Orbiter is designed to provide a deeper understanding of how the Sun creates and controls the heliosphere, the vast bubble of charged particles blown by solar wind into the surrounding galaxy.
  2. Link Solar Outputs to Solar Wind: It aims to closely link in-situ measurements of the solar wind with the observations of the Sun’s surface and outer atmosphere, helping scientists understand the source regions of solar wind and solar energetic particles.
  3. Study of Solar Polar Regions: It includes observing the Sun’s poles, providing the first-ever images of these regions, which are crucial for understanding the Sun’s magnetic field and its variations over time.

Technical and Design Features:

  • Orbit: Solar Orbiter’s path takes it within approximately 42 million kilometers (about 26 million miles) of the Sun, closer than the orbit of Mercury. The orbit is highly elliptical and gradually increases in inclination to the ecliptic, eventually reaching angles of more than 30 degrees, allowing for unique views of the Sun’s poles.
  • Heat Shield: The spacecraft is equipped with a heat shield to protect it from the intense solar radiation, enabling it to withstand temperatures up to 500 degrees Celsius (about 932 degrees Fahrenheit).
  • Instrumentation: It carries a suite of ten science instruments:
    • Six remote-sensing instruments to image the solar atmosphere and measure the solar output.
    • Four in-situ instruments to sample particles and fields in the vicinity of the spacecraft.

Scientific Instruments and Their Roles:

  • Remote Sensing Instruments: These include imagers and spectrometers that study the solar atmosphere, observing the Sun’s corona, solar flares, and detecting eruptions of solar material and how they propagate.
  • In-Situ Instruments: These measure the composition, timing, and characteristics of solar wind particles and the interplanetary magnetic field.

Achievements and Goals:

  • Holistic Approach: By combining remote sensing and in-situ measurements, Solar Orbiter provides a comprehensive picture of how magnetic fields and material ejected from the Sun give rise to the heliosphere.
  • Mission Progress: As of its launch and early operations, Solar Orbiter has already begun to send back data and images, including detailed views of the Sun’s surface and solar phenomena such as coronal loops and potential coronal mass ejection source regions.

The Solar Orbiter mission is designed to last for about 10 years, with the potential to fundamentally change our understanding of the Sun’s behavior and its effects on the space environment between the Sun and Earth. This mission complements other solar missions by providing data from perspectives that are not possible with other spacecraft, offering new insights into the Sun’s dynamics and its influence on the broader solar system.

Daniel K. Inouye Solar Telescope (DKIST)

The Daniel K. Inouye Solar Telescope (DKIST), formerly known as the Advanced Technology Solar Telescope (ATST), is the world’s largest solar telescope, situated at the Haleakalā Observatory on Maui, Hawaii. This state-of-the-art facility, which began operations in 2020, is designed to provide unprecedented insights into the Sun’s atmosphere and dynamics by capturing high-resolution images and offering detailed spectroscopic information.

Key Features and Capabilities:

  • Aperture Size: DKIST boasts a 4-meter (13 feet) diameter primary mirror, the largest for a solar telescope, which allows for exceptionally fine details to be observed on the Sun’s surface, down to 20 kilometers in size.
  • Location: Situated at over 3,000 meters (about 10,000 feet) above sea level on Haleakalā, the site benefits from stable atmospheric conditions which are crucial for high-resolution solar observations.
  • Heat Management: To manage the intense heat from focused sunlight, DKIST uses a sophisticated cooling system and a heat stop that blocks out most of the sunlight, allowing only a small portion to reach the scientific instruments.

Scientific Instruments:

DKIST is equipped with a suite of cutting-edge instruments designed to study various aspects of solar physics: - Visible Broadband Imager (VBI): Captures high-resolution images of the solar photosphere and chromosphere. - Visible Spectro-Polarimeter (ViSP): Analyzes the magnetic fields of the Sun by measuring the polarization of light at different wavelengths. - Visible Tunable Filter (VTF): Provides high-resolution imaging spectroscopy and spectropolarimetry in the visible light spectrum, crucial for understanding magnetic fields and dynamic events on the Sun. - Diffraction-Limited Near-Infrared Spectro-Polarimeter (DL-NIRSP): Measures the magnetic fields in the Sun’s chromosphere using near-infrared wavelengths. - Cryogenic Near-Infrared Spectro-Polarimeter (Cryo-NIRSP): Observes the Sun’s corona outside of solar eclipses, examining the magnetic fields in the outer solar atmosphere.

Objectives and Scientific Goals:

  • Magnetic Field Studies: DKIST aims to directly measure and understand the complex magnetic fields of the Sun, including their role in events like solar flares and coronal mass ejections.
  • Solar Atmosphere Dynamics: It explores the dynamics of the solar atmosphere across different layers, from the photosphere to the corona.
  • Energy Transfer: The telescope investigates how energy is transferred and converted in the solar atmosphere, particularly how magnetic energy leads to solar eruptions and influences space weather.

Impact and Importance:

  • High Resolution: DKIST’s high-resolution capabilities allow for detailed studies of fine-scale solar structures and processes that were not possible before, potentially leading to breakthroughs in our understanding of the Sun.
  • Space Weather Forecasting: Improved knowledge of solar magnetic activity and its manifestations can enhance space weather forecasting, which is crucial for protecting satellites, astronauts, and electrical grids on Earth.

Overall, DKIST is a flagship facility in solar science, providing critical data that will enable scientists to delve deeper into the fundamental processes at work on the Sun, enhancing our understanding of the star that has the most profound influence on our solar system.

Deets

Stars

  • Andrew King
  • Volume 322
  • Published 2012

The Sun

  • The Sun: Philip Judge
  • Volume 638
  • Published 2020