From 4194ff41da35226fa39a0f9d394b04519b3b3d0f Mon Sep 17 00:00:00 2001 From: Lorenzo <140400600+lollocava00@users.noreply.github.com> Date: Tue, 28 May 2024 10:21:34 +0200 Subject: [PATCH] Update README.md --- README.md | 38 +++++++++++++++++++------------------- 1 file changed, 19 insertions(+), 19 deletions(-) diff --git a/README.md b/README.md index 87cf302..2e37ea0 100644 --- a/README.md +++ b/README.md @@ -1,5 +1,5 @@ # SNR Evolution simulation -The results presented here are based on a simplified version of the ["ZEUS-2D"](https://articles.adsabs.harvard.edu/pdf/1992ApJS...80..753S) code in an attempt to simulate the evolution of an expanding Supernova Remnant (SNR), both when we neglect or take into account radiative losses. +The results presented here are based on a simplified version of the ["ZEUS-2D"](https://articles.adsabs.harvard.edu/pdf/1992ApJS...80..753S) code in an attempt to simulate the evolution of an expanding **Supernova Remnant (SNR)**, both when we neglect or take into account radiative losses. I used a one dimensional version of Zeus-2D to reproduce the evolution of a Supernova explosion which is assumed to be spherically symmetric. The numerical approach taken is discussed in the following chapter. @@ -23,16 +23,16 @@ $$p=(\gamma- 1)\epsilon$$ where $\gamma$ is set to the value $\gamma=\frac{5}{3}$. -Equations 1-3 constitute a set of hyperbolic partial differential equations, which are tackled in the code through a finite-differences approach (upwind I); in particular, the code breaks the solving procedure in two separate steps: -- Source step, where only the soure terms of the equations are considered: +Equations 1-3 constitute a set of hyperbolic partial differential equations, which are tackled in the code through a **finite-differences approach (upwind I)**; in particular, the code breaks the solving procedure in two separate steps: +- **Source step**, where only the soure terms of the equations are considered: $$\rho \frac{Dv}{Dt} =-\nabla p -\nabla Q $$ $$ \frac{D\epsilon}{Dt}=-p\nabla \cdot v -Q \nabla \cdot v$$ - where Q is an artificial viscosity term that will be discussed later; + where **Q is an artificial viscosity** term that will be discussed later; - - Transport step, which accounts for the effects of fluid advection: + - **Transport step**, which accounts for the effects of fluid advection: $$ \frac{\partial }{\partial t} \int_V \rho dV =-\int_{\partial V} \rho v \;dS $$ @@ -40,8 +40,8 @@ $$ \frac{\partial}{\partial t} \int_V \rho v dV =-\int_{\partial V} \rho v^2 \; $$\frac{\partial}{\partial t} \int_V \epsilon dV =-\int_{\partial V} \epsilon v \;dS $$ - (the grid velocity is assumed to be null for our purposes). -The aforementioned equations are solved on two staggered grids, named $x_a$ and $x_b$, where $x_b(i)=x_a(i+\frac{1}{2})$. A choice is also available between two differente coordinate systems (cartesian or spherical). +(the grid velocity is assumed to be null for our purposes). +The aforementioned equations are solved on two *staggered grids*, named $x_a$ and $x_b$, where $x_b(i)=x_a(i+\frac{1}{2})$. A choice is also available between two differente coordinate systems (cartesian or spherical). ![Alt text](plots/shocktube.png?raw=true) @@ -49,11 +49,11 @@ The aforementioned equations are solved on two staggered grids, named $x_a$ and On average, a total energy of about $10^{51} erg$ is injected into the interstellar medium (ISM) by supernovae. The resulting shock wave propagates outwards at very high speed, but quickly loses large amounts of energy due to compression and heating of the ISM and also because of radiative losses. To allow the modelling of this rather complex process, some simplifying assumptions were made, which include radial symmetry of the entire system, a uniform ISM, the absence of a stellar wind phase before the explosion and thermal conduction is neglected. -The code is set up to allow for the analysis of an expanding SuperNova Remnant (SNR), more specifically in the following terms: +The code is set up to allow for the analysis of an expanding **Supernova Remnant (SNR)**, more specifically in the following terms: - evolution of $\rho$, $p$ , $T$ and $v$ over various time ranges; -- evolution of the shock radius generated by the supernova event in radial expansion; -- evolution of the X-RAY luminosity of the source as a function of time; -- the kinetic and thermal energy fractions injected into the surrounding ISM as a function of time. +- evolution of the **shock radius** generated by the supernova event in radial expansion; +- evolution of the **X-RAY luminosity** of the source as a function of time; +- the **kinetic and thermal energy** fractions injected into the surrounding ISM as a function of time. The aforementioned characteristics are evaluated both in the presence and lack of radiative cooling as to highlight possible differences. The initial conditions chosen for this setting are typical values in the ISM: @@ -70,7 +70,7 @@ The central energy values are set to $E_0 = 10^{51} \, erg$ in order to simulate $$ e(2)=e(3)=\frac{E_0}{\frac{4}{3}\pi x_a^3} $$ -The time step over which the DO loop runs is set to: +The time step over which the *DO loop* runs is set to: $$ \Delta t=cfl *\frac{\Delta x}{|v|+c_s}$$ @@ -83,7 +83,7 @@ Such values for $\Delta t$ always satisfy the stability condition of Upwind I or $$ \Delta t \leq \frac{\Delta x}{v}$$ ## Results without cooling -The DO loop is performed over a selection of time spans: +The *DO loop* is performed over a selection of time spans: $$ t= 2\cdot 10^4,4\cdot 10^4,6\cdot 10^4,8\cdot 10^4, 10^5, 2\cdot 10^5,3\cdot 10^5, 4\cdot 10^5,5\cdot 10^5 \, yrs $$ @@ -105,8 +105,8 @@ and drops right after $R_{shock}$. The same is true for figure 3, where all the quantities keep evolving following the same trends. ## Results with cooling -The contribution of a cooling function $\Lambda(T)$ is added; $\Lambda(T)$ defines the bolometric power emitted by the surrounding -medium in the form of thermal radiation and takes into account different properties of the plasma. The function I used was first introduced by Sutherland and Dupita: +The contribution of a **cooling function** $\Lambda(T)$ is added; $\Lambda(T)$ defines the bolometric power emitted by the surrounding +medium in the form of thermal radiation and takes into account different properties of the plasma. The function I used was first introduced by *Sutherland and Dupita*: $$\Lambda(T)=10^{-22}(8.6\cdot 10^{-3}T_{KeV}^{-1.7}+0.058T_{KeV}^{0.5}+0.063 \quad for \quad T>0.02 \,KeV $$ @@ -126,16 +126,16 @@ Another observation that can be made with regards to the density is that, after Another observation that can be made with regards to the temperature is that after about $2\cdot 10^4 yrs$ the shell temperature (look at shock radius) is not decreasing, this signifies the end of the adiabatic phase and the beginning of the radiative phase in which the energy that is gained by impact with the ISM is instantly radiated away. ## Shock radius -The shock radius describes the distance from the centre of the explosion to the ”front” of the shock wave. An analytical expression exists, called themSedov solution. It allows the calculation of the shock radius via +The shock radius describes the distance from the centre of the explosion to the ”front” of the shock wave. An analytical expression exists, called the **Sedov solution**. It allows the calculation of the shock radius via $$ R_s=\left( \frac{2E_0}{\rho_0} \right)^{1/5}t^{2/5}.$$ -However, this solution is only relevant in the adiabatic phase of the SNR expansion and should stop working after radiative losses become important. +However, this solution is only relevant in the **adiabatic phase** of the SNR expansion and should stop working after radiative losses become important. Numerically the front of the shock wave can not be defined in a rigorous way. In order to obtain a fixed value for the shock radius it was chosen as the point in space at which the density is maximal. The code computes the value of the shock radius every $10^3$ years by introducing an IF statement near the end of the DO loop; the right time step is selected by using a counter that only contains multiples of $10^3$. -Inside the IF statement the value of the Shock radius is taken as the grid point that contains the maximum value of the density. This is done by using the function maxval(d). +Inside the IF statement the value of the Shock radius is taken as the grid point that contains the maximum value of the density. This is done by using the function *maxval(d)*. As the literature on Supernovae tells us, radiative losses start to beacome important after a few $10^4 yrs$ strongly depending on the ISM density. Figure 7 shows that in our case this happens at about $log(t) \sim 4.5$ or $3\cdot 10^4 yrs$ . @@ -151,7 +151,7 @@ are plotted in Figure 7. ![Alt text](plots/R_shock2.png?raw=true) ## X-Ray luminosity -The supernova remnant is a source of X-RAY radiation, mainly caused by bremsstrahlung emission, recombination continuum and two-photon emission; the values initially increase, but as the energy of the charged particles is irradiated through high energy photons over time the luminosity decreases. The following plot shows the evolution of the source’s X-RAY luminosity both in the presence of a cooling function and in the absence of it. One can notice how the more efficient energy dissipation caused by $\Lambda$ causes a steeper decrease over time. +The supernova remnant is a source of X-RAY radiation, mainly caused by **bremsstrahlung emission**, recombination continuum and two-photon emission; the values initially increase, but as the energy of the charged particles is irradiated through high energy photons over time the luminosity decreases. The following plot shows the evolution of the source’s X-RAY luminosity both in the presence of a cooling function and in the absence of it. One can notice how the more efficient energy dissipation caused by $\Lambda$ causes a steeper decrease over time. ![Alt text](plots/LX1.png?raw=true)