finished Stellar pulsations

Chapter_14/stellar-pulsation.ipynb

@@ -398,7 +398,7 @@
     "\n",
     "Note that $\\delta P$ varies in time, while $P_o$ is a constant.  When the variables written in this manner are inserted into the differential equations, the terms containing only equilibrium quantities cancel, and terms that involve powers greater than second order (i.e., greater than $(\\delta P)^2)$ may be discarded because they are negligibly small.  The resulting linearized differential equations and their associated boundary conditions (also linearized) are similar to the equations for a wave on a string or in an organ pipe.\n",
     "\n",
-    "Only certain standing waves with specific periods are permitted, where the the pulsation modes of the star are clearly identified.  The equations are sufficiently complicated that the motion of the star is forced to be sinusoidal, and the limiting value of the pulsation amplitude cannot be determined.  Modeling the complexities of the full nonlinear behavior of the stellar model is thus sacrificed."
+    "Only certain standing waves with specific periods are permitted, where the pulsation modes of the star are clearly identified.  The equations are sufficiently complicated that the motion of the star is forced to be sinusoidal, and the limiting value of the pulsation amplitude cannot be determined.  Modeling the complexities of the full nonlinear behavior of the stellar model is thus sacrificed."
    ]
   },
   {
@@ -603,7 +603,7 @@
     "where the $K^{\\pm m}_\\ell$ are normalization constants and $i$ is the imaginary number.\n",
     "\n",
     "```{note}\n",
-    "Recall from Euler's formula that $e^{\\pm mi\\phi} = \\cos(m\\phi) \\pm i\\sin{m\\phi}$.  thus the real part of $e^{\\pm mi\\phi}$ is just $\\cos(m\\phi)$.\n",
+    "Recall from Euler's formula that $e^{\\pm mi\\phi} = \\cos(m\\phi) \\pm i\\sin{m\\phi}$.  Thus the real part of $e^{\\pm mi\\phi}$ is just $\\cos(m\\phi)$.\n",
     "```\n",
     "\n",
     "The patterns for nonzero $m$ represent *traveling* waves that move across the star parallel to its equator.  The time required for the waves to travel around the star is $|m|$ times the star's pulsation period.  \n",
@@ -666,24 +666,104 @@
     "\n",
     "### The Brunt-Väisälä (Buoyancy) Frequency \n",
     "\n",
-    "### The $g$ and $p$ Modes as Probes of Stellar Structure"
+    "To better understand the $g$-modes, **consider a small bubble of stellar material that is displaced upward from its equilibrium position by an amount $dr$.**  We wll assume that the motion occurs\n",
+    "\n",
+    "1. slow enough for the pressure in the bubble, $P^b$, is always equal to the pressure of its surroundings, $P^s$; and\n",
+    "\n",
+    "2. rapid enough so that no heat is exchanged between the bubble an dits surroundings (i.e., adiabatic).\n",
+    "\n",
+    "If the density of the displaced bubble is greater than the density of its new surroundings, the bubble will fall back to its original position.  The *net restoring force per unit volume* on the bubble is the difference between the upward buoyant force (i.e., Archimedes' law) and the downward gravitational force at its **f**inal location:\n",
+    "\n",
+    "$$ f_{\\rm net} = \\left( \\rho_f^s - \\rho_f^s\\right)\\ g, $$\n",
+    "\n",
+    "where $g$ represents the local gravity ($GM_r/r^2$).  Using a Taylor expansion for the densitites about their initial positions in \n",
+    "\n",
+    "$$f_{\\rm net} = \\left[\\left( \\rho_i^s + \\frac{d\\rho^s}{dr}dr\\right) - \\left( \\rho_i^b + \\frac{d\\rho^b}{dr}dr\\right)\\right]\\ g. $$\n",
+    "\n",
+    "The initial densities of the bubble and its surroundings are the same ($\\rho_i^s = \\rho_i^b$) and after canceling terms, leaves\n",
+    "\n",
+    "$$f_{\\rm net} = \\left(\\frac{d\\rho^s}{dr} - \\frac{d\\rho^b}{dr}\\right)\\ g\\ dr. $$\n",
+    "\n",
+    "Because the motion of the bubble is adiabatic (and considering the [conditions for convection](https://saturnaxis.github.io/ModernAstro/Chapter_11/interiors-of-stars.html#a-criterion-for-stellar-convection)), the $d\\rho^b/dr$ can be replaced to get\n",
+    "\n",
+    "$$ f_{\\rm net} = \\left(\\frac{d\\rho^s}{dr} - \\frac{\\rho^b}{\\gamma P_i^b}\\frac{dP^b}{dr}\\right)\\ g\\ dr. $$\n",
+    "\n",
+    "Actually, all of the $b$ superscripts can be changed to $s$ because the initial densities are equal.  Using our first assumption, the pressures inside **and** outside the bubble are *always* the same.  Thus all quantities in refer to the stellar material surrounding the bubble, which allows us to drop the subscripts/superscipts and results in \n",
+    "\n",
+    "```{math}\n",
+    ":label: net_bubble_force\n",
+    "f_{\\rm net} &= \\left(\\frac{d\\rho}{dr} - \\frac{\\rho}{\\gamma P}\\frac{dP}{dr}\\right)\\ g\\ dr\\\\\n",
+    "&= A \\rho g\\ dr.\n",
+    "```\n",
+    "\n",
+    "The $A$ represents the difference in the rate of change for the pressure and density (see Eq. {eq}`press_sound_speed`).\n",
+    "\n",
+    "If $A>0$, the net force has the same sign as $dr$ and the bubble will contiune to move away from its equilibrium position.  This is the condition necessary for *convection* to occur.\n",
+    "\n",
+    "However, if $A<0$, then the net force will be in an opposite direction relative to the displacement and the bubble will be pushed back toward its equilibrium position.  In this case, the result of Eq. {eq}`net_bubble_force` has the form of Hooke's law with the restoring force proportional to the displacement.  We should expect that the bubble will *oscillate* about its equilibrium position with simple harmonic motion when $A<0$.\n",
+    "\n",
+    "Dividing the force per unit volume $f_{\\rm net}$ by the mass per unit volume $\\rho$, gives the force per unit mass (or acceleration): $a = f_{\\rm net}/\\rho = Ag\\ dr$.  As a result, we have\n",
+    "\n",
+    "$$ a = -N^2 dr = Ag\\ dr, $$\n",
+    "\n",
+    "where $N$ represents the angular frequency of the bubble about its equilibrium position and is called the **Brunt-V&auml;is&auml;l&auml; frequency$$ or the **bouyant frequency**.  More explicitly,\n",
+    "\n",
+    "\\begin{align}\n",
+    "N &= \\sqrt{-Ag}, \\\\\n",
+    "&= \\sqrt{\\frac{\\rho}{\\gamma P}\\frac{dP}{dr} - \\left(\\frac{d\\rho}{dr}\\right)\\ g}.\n",
+    "\\end{align}\n",
+    "\n",
+    "The bouyancy frequency is zero at the stellar center (where $g=0$) and at the edges of the convection zones (where $A=0$).\n",
+    "\n",
+    "### The $g$ and $p$ Modes as Probes of Stellar Structure\n",
+    "\n",
+    "The *sloshing* effect of neighboring regions of the star produces the internal gravity waves that are responsible fo rthe $g$-modes of a nonradially pulsating star.  The frequency of a $g$-mode is determined by the value of $N$ averaged across the star.\n",
+    "\n",
+    "$f$-modes and $g$-modes each produce different profiles for how material moves radially within a star.  This makes them useful to astronomers attempting to study stellar interiors.  The $g$-modes involve significant movement of the stellar material deep within the star, while the $p$-mode's motion are confined to the stellar surface.  Thus, $g$-modes provide a view into the very heart of a star, while $p$-modes present the conditions in the surface layers.\n",
+    "\n",
+    "```{figure-md} nonradial-gmodes-fig\n",
+    "<img src=\"Figure18.jpg\" alt=\"Nonradial g-modes\"  width=\"600px\">\n",
+    "\n",
+    "Nonradial $g$-modes with $\\ell = 2$.  The waveforms have been arbitrarily scales so that $\\delta r/R = 1$ a the star's surface.   Figure credit: Carroll & Ostlie (2007).\n",
+    "```"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "## Helioseismology and Asteroseismology\n",
+    "Nonradial pulsation is a the heart of a science called [helioseismology](https://en.wikipedia.org/wiki/Helioseismology).  A typical solar oscillation mode has a very low amplituded, with a surface velocity of only $10\\ {\\rm cm/s}$ or less, and a luminosity variation $\\delta L/L_\\odot$ of only $10^{-6}$.  With an incoherent superposition of roughly *ten million* modes rippling through the Sun's surface and interior, our star is \"ringing\" like a bell.\n",
     "\n",
     "### The Five-Minute Solar Oscillations \n",
+    "The oscillations observed on the Sun have modes with periods from $3-8$ minutes and very short horizontal wavelengths ($\\ell$ ranging up to 1000 or more).  These so-called *five-minute oscillations* are $p$-modes.  The five-minute $p$-modes are concentrated below the photosphere within the Sun's convection zone, where $g$-modes are located deep in the solar interior.  By studying these $p$-mode oscillations, astronomers have been able to gain new insights in the structure of the Sun in these regions.\n",
     "\n",
     "### Differential Rotation and the Solar Convection Zone\n",
     "\n",
+    "Based on studies of helioseismology (combined with detailed stellar evolution calculations), we know the base of the solar convection zone lies at $0.714\\ R_\\odot$ with a temperature of about $2.18 \\times 10^6\\ {\\rm K}$.  The rotational splitting observed for $p$-mode frequencies indicates that differential roation observed at the Sun' surface decreases slightly down through the convection zone.\n",
+    "\n",
+    "Those $p$-modes with shorter horizontal wavelengths (larger $\\ell$) do not penetrate the convection zone as deeply, so the difference in rotational frequency splitting with $\\ell$ reveals the depth dependence of the rotation.  The measured depth dependence of the differential rotation comes from the dependence on the rotational frequency splitting on $m$.\n",
+    "\n",
+    "Below the convection zone, the equatorial and polar rotation rates converge to a single value at $r/R_\\odot \\approx 0.65$.  To convert the Sun's magnetic field from a poloidal to a toroidal geometry a change in the rotation rate with depth is needed.  The Sun's magnetic dynamo is probably seated in the tachocline at the interface between the radiation and convection zones.\n",
+    "\n",
     "### Probing the Deep Interior\n",
+    "Astronomers had more difficulty using the solar $g$-modes as a probe of the Sun's interior because they dwell beneath the convection zone, where their amplitudes are significantly diminished at the Sun's surface.  Here is an [astrobites](https://astrobites.org/2018/02/07/the-deepest-rumblings-of-the-sun/) article summarizing a recent attempt to measure the $g$-modes indirectly.  However, a definitive measurement remains hotly debated.\n",
     "\n",
     "### Driving Solar Oscillations\n",
     "\n",
-    "### $\\delta$ Scuti Stars and Rapidly Oscillating Ap Stars"
+    "The question of the mechanism responsible for driving the solar oscillations has not been conclusevely answered.  Our main-sequence Sun lies beyond the red edge of the instability strip on the H-R diagram.  Therefore, Eddington's valve mechanism cannot be responsible for the solar oscillations.  However, the timescale for convection near the top of the convection zone is a few minutes, and it is strongly suspected that the $p$-modes are driven by tapping in to the turbulent energy of the convection zone itself, where the $p$-modes are confined.\n",
+    "\n",
+    "### $\\delta$ Scuti Stars and Rapidly Oscillating Ap Stars\n",
+    "\n",
+    "The techniques of helioseismology can be applied to other stars to become [asteroseismology](https://en.wikipedia.org/wiki/Asteroseismology), where astronomers study the pulastion modes of other stars in order to investigate their internal structures, chemical compositions, rotation, and magnetic fields.\n",
+    "\n",
+    "$\\delta$ Scuti stars are Pop I main-sequence and giant stars within the spectral type range $\\rm A-F$. They tend to pulsate in low-overtone radial modes and in low-order $p$-modes (and possibly $g$-modes).  The amplitudes of $\\delta$ Scutis are fairly small, ranging from a few millimagnitudes to roughly 0.8 mag.  Population II subgiants also exhibit radial and nonradial oscillations, which are known as SX Phoenicis stars.\n",
+    "\n",
+    "The **rapidly oscillating Ap stars** ($\\rm roAp$) are found in the same portion of the H-R diagram as the $\\delta$ Scuti stars.  These stars have peculiar surface chemical compositions (hence the \"p\" designation), are rotating, and have strong magnetic fields.  The unusual chemical composition is likely du to the settling of heavier elements, similar to the elmenetal diffusion that has occured near the surface of the Sun.\n",
+    "\n",
+    "Some elements may have been eleveated in the atmosphere if they have a significant number of absorption lines near the peak of the star's blackbody spectrum.  These atoms preferentially absorb photons that impart a net upward momentum.  If the atmosphere is sufficiently stable against turbulent motions, some of these atoms will tend to drift upward.\n",
+    "\n",
+    "$\\rm roAp$ stars have very small pulsation amplitutdes of less than 0.016 mag.  It appears that they primarly pulsate in higher-order $p$-modes and that the axis for the pulsation is aligned with the magnetic field axis, which is tilted relative to the roation axis.  $\\rm roAp$ star are among the most well-studied of main-sequene stars other than the Sun, but the pulsation driving mechanism still remains in question."
    ]
   },
   {