Robert Jelle Rutten (1942 - 2022)

So long, my teacher and friend. You will be hugely missed.

1D Solar Atmosphere Models in IDL: Standard Solar Model (from "The Sun", by Stix)

Stix, in his book ("The Sun: An Introduction", Springer, 2002) in Sect.2.4 introduces a standard solar model from the core to the surface defined as $\tau = 2/3$. His Table 2.4 (on p.56) lists the values of various quantities of this model versus the column mass or the height. Check the book for the details of the model. The table in ascii format is here: stix_tab.2.4.txt The columns are: $m/m_\odot$, $r/r_\odot$, $p \mathrm[Pa]$, $T \mathrm[K]$, $\rho \mathrm[kg/m^3]$, $L/L_\odot$, $X$, $\mu$, $\Gamma_1$.

NGC 7319: GTC vs JWST

The first pictures from James Webb Space Telescope are simply mind blowing! Here is a comparison of NGC 7319 (from Stephan's quintet of galaxies) from Gran Telescopio Canarias (still the largest single aperture optical telescope) and from JWST. One should keep in mind that these are different instruments observing and different wavelengths. The alignment is also not perfect.

Quadratic splines: Part I

Some basic properties of parabola in the context of the second-order Hermite interpolation equation. 

There is a one-dimensional dataset with $n$ points: $x = (x_0, \dots, x_{n-1})$ and $y = (y_0, \dots, y_{n-1})$, such that $x_{i} > x_{i-1}$ for any $i$. We introduce the following labels: $$h_{i-1} = x_i - x_{i-1}$$ $$\delta_{i-1} = \frac{y_i - y_{i-1}}{x_i - x_{i-1}}$$ $h$ is always positive, while $\delta$ can have any value. On each segment, we approximate the unknown function with a parabola. On the $[i-1, i]$ segment: \begin{equation}f(x) = a_0 + a_1 (x - x_{i-1}) + a_2 (x - x_{i-1})^2\end{equation} The first and the second derivative of the parabola are: \begin{equation}f'(x) = a_1 + 2 a_2 (x - x_{i-1})\end{equation} \begin{equation}f''(x) = 2 a_2 \end{equation} The Hermite interpolation theorem provides us with a tool to determine the $a$ coefficients on the $[i-1, i]$ segment if we know $y_{i-1}$, $y_i$ and one of the first derivative in the end points, either $y'_{i-1}$ or $y'_i$. Let's assume that we know ("know" in the sense "we can compute") $y'_{i-1}$. Then there are three equations that the parabola must obey: \begin{equation}f(x_{i-1}) = y_{i-1} = a_0\end{equation} \begin{equation}f'(x_{i-1}) = y'_{i-1} = a_1\end{equation} \begin{equation}f(x_{i}) = y_i = a_0 + a_1 h_{i-1} + a_2 h_{i-1}^2\end{equation} The solution is then trivial: \begin{equation}a_0 = y_{i-1}\end{equation} \begin{equation}a_1 = y'_{i-1}\end{equation} \begin{equation}a_2 = \frac{y_{i} - y_{i-1} - y'_{i-1} h_{i-1}}{h_{i-1}^2}\end{equation} Once the $a$ coefficients are known, the parabola is fully determined. That also means that the derivative in the point $i$ is fixed: $$f'(x_{i}) = y'_i = a_1 + 2 a_2 (x_i - x_{i-1}) = y'_{i-1} + 2 \frac{y_{i} - y_{i-1} - y'_{i-1} h_{i-1}}{h_{i-1}^2} h_{i-1}$$ $$y'_i = \frac{2(y_{i} - y_{i-1}) - y'_{i-1} h_{i-1}}{h_{i-1}} $$ \begin{equation}y'_i = 2\delta_{i-1} - y'_{i-1}\end{equation} Local extremum

The extremum of this parabola is located at $x_m$: $$f'(x_m) = 0 = a_1 + 2 a_2 (x_m - x_{i-1})$$ \begin{equation}x_m = x_{i-1} - \frac{a_1}{2a_2}\end{equation} Monotonicity

The function $f(x)$ is monotone on the $[i-1, i]$ segment if it does not have an extremum on it, i.e. if $$x_m < x_{i-1}\;\;\;\;\;\;\;\;\;\;\mathrm{or}\;\;\;\;\;\;\;\;\;\; x_m > x_i$$ The first condition readily translates into: $$x_{i-1} - \frac{a_1}{2a_2} < x_{i-1}$$ $$ \frac{a_1}{2a_2} > 0$$ or, when we substitute $a_1$ and $a_2$: $$ \frac{y'_{i-1} h_{i-1}^2}{2(y_{i} - y_{i-1} - y'_{i-1} h_{i-1})} > 0$$ $$ \frac{y'_{i-1} h_{i-1}}{2(\delta_{i-1} - y'_{i-1})} > 0$$ As by definition $h_{i-1} > 0$ $$ \frac{y'_{i-1}}{2(\delta_{i-1} - y'_{i-1})} > 0$$ If $y'_{i-1} > 0$ then $\delta_{i-1} - y'_{i-1} > 0$ and $\delta_{i-1} > y'_{i-1}$. If $y'_{i-1} < 0$, then $\delta_{i-1} < y'_{i-1}$. Therefore, the first condition ($x_m < x_{i-1}$) reduces to: \begin{equation}\boxed{|\delta_{i-1}| > |y'_{i-1}|}\end{equation} The second condition leads to: $$x_{i-1} - \frac{a_1}{2a_2} > x_{i}$$ $$-x_{i} + x_{i-1} - \frac{a_1}{2a_2} > 0$$ $$h_{i-1} + \frac{a_1}{2a_2} < 0$$ $$h_{i-1} + \frac{y'_{i-1} h_{i-1}}{2(\delta_{i-1} - y'_{i-1})} < 0$$ $$\frac{2\delta_{i-1} - y'_{i-1}}{2(\delta_{i-1} - y'_{i-1})} < 0$$ There are two cases. First, if $2\delta_{i-1} - y'_{i-1} > 0$, i.e. $2\delta_{i-1} > y'_{i-1}$. then it must be $\delta_{i-1} - y'_{i-1}< 0$, i.e.$\delta_{i-1} < y'_{i-1}$. Therefore, the first case leads to $2\delta_{i-1} > y'_{i-1} > \delta_{i-1}$. Secondly, if $2\delta_{i-1} - y'_{i-1} < 0$, i.e. $2\delta_{i-1} < y'_{i-1}$, it must be $\delta_{i-1} - y'_{i-1}> 0$, i.e. $\delta_{i-1} > y'_{i-1}$. So the second case leades to $\delta_{i-1} > y'_{i-1}> 2\delta_{i-1}$. The first case is possible only when both $\delta_{i-1}$ and $y'_{i-1}$ are positive and the second case is possible only when both are negative. Therefore, together, the two cases reduce the second condition ($x_m > x_{i}$) to: \begin{equation}\boxed{|2\delta_{i-1}| > |y'_{i-1}| > |\delta_{i-1}|}\end{equation} Together, the two conditions combine into \begin{equation}\boxed{|2\delta_{i-1}| > |y'_{i-1}|}\end{equation} A much faster way to reach the same result is to realize that the the derivatives $y'_{i-1}$ and $y'_{i}$ must be of the same sign if the function is monotone on the $[i-1, i]$ segment, i.e. $$y'_{i-1} \cdot (2\delta_{i-1} - y'_{i-1}) > 0$$ which obviously leads to $|2\delta_{i-1}| > |y'_{i-1}|$.

Therefore, for a Hermite parabola defined by $y_{i-1}$, $y_i$ and $y'_{i-1}$ to be monotone on the interval $(i-1, i)$, the derivative $y'_{i-1}$ must be of the same sign as the linear slope $\delta_{i-1}$ and smaller in magnitude than $2\delta_{i-1}$.

Convexity

For the function $f$ to be convex on $(i-1, i)$ (if parabola is convex on a segment, it's convex everywhere) the condition is $f''(x) > 0$. In the case of our parabola: $$f''(x) = 2a_2 > 0$$. $$\frac{y_{i} - y_{i-1} - y'_{i-1} h_{i-1}}{h_{i-1}^2} > 0$$ or $$y_{i} - y_{i-1} - y'_{i-1} h_{i-1} > 0$$ \begin{equation}\delta_{i-1} > y'_{i-1}\end{equation} This result is obvious $\delta_{i-1} = y'_{i-1}$ is the straight line going through the end-points of the segment. For the parabola to be concave, it must hold $f''(x) < 0$, i.e. \begin{equation}\delta_{i-1} < y'_{i-1}\end{equation} 

 

XXXI Canary Islands Winter School: Computational Fluid Dynamics in Astrophysics (Photos)

Two weeks of interesting lectures, intensive hands-on exercises, discussions and meeting new people... A couple of photos by Claudio Dalla Vecchia that perfectly capture the spirit of the school. 

XXXI Canary Islands Winter School: Computational Fluid Dynamics in Astrophysics

Registration is now open for the XXXI Canary Islands Winter School of Astrophysics, which will be held this year in La Laguna, Tenerife, Spain from 19-28 November, on the topic of "Computational Fluid Dynamics in Astrophysics". Confirmed invited lecturers are

Fernando Moreno-Insertis (Fluid dynamics, conservation laws and waves)
Ake Nordlund (Stellar and planetary formation)
Maarit Käpylä (Dynamics of solar and stellar convection zones)
Matthias Rempel (Solar Atmosphere)
Sascha Husa (General relativity and compact objects)
Stefanie Walch-Gassner (Interstellar matter)
Tom Theus (Galaxy evolution)

The school will consist of lectures, seminars tutorials, and social events including visits to the observatories in Tenerife and La Palma. It is aimed at advanced graduate students in astrophysics, as well as postdoctoral researchers, with a strong interest in the computational fluid dynamics and it applications astrophysics.

For further information and to register, please visit

http://www.iac.es/winterschool/2019/

Claudio Dalla Vecchia
Nikola Vitas
Elena Khomenko

Three-dimensional simulations of solar magneto-convection including effects of partial ionization

Astronomy & Astrophysics Volume 618 (October 2018) is just out with a figure from our paper on the cover!

The paper (Khomenko, Vitas, Collados & de Vicente 2018: A&A, ADS) describes the effects of the partial ionization on the structure, dynamics and energy balance of the low chromosphere.