## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|ASTY221

2022年12月28日

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## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|First Order Differential Equations

A differential equation of first order determines a time-dependent function $x(t)$ by a relation between the function and its first derivative $\dot{x}$. An example is the equation for radioactive decay:
$$\dot{x}=\lambda x,$$

The solution of this equation is $x(t)=x_0 \mathrm{e}^{-\lambda t}$, where $x_0=x(0)$ is said to be the initial value. While it is straightforward to solve a linear differential equation, in which all terms are linear in $x$ or $\dot{x}$, non-linear differential equations are more challenging.
Consider the Bernoulli equation ${ }^1$ :
$$\dot{x}=\alpha(t) x+\beta(t) x^\rho,$$
where $\alpha(t)$ and $\beta(t)$ are given functions and $\rho$ is a real number. It encompasses important differential equations as special cases, for example, Eq. (4.1) follows for $\alpha(t)=\lambda, \beta(t)=0, \rho=0$ and the logistic equation describing population dynamics for $\alpha(t)=a, \beta(t)=b$ with constants $a>0, b<0$, and $\rho=2$ (see Exercise 4.1). From the viewpoint of numerics, Bernoulli-type equations are interesting because an analytic solution is known and can be compared to numerical approximations.
In the following, we will attempt to numerically solve an example for a Bernoulli equation from astrophysics, namely the equation for the radial expansion of a socalled Strömgren sphere. When a hot, massive star is borne, it floods its surroundings with strongly ionizing UV radiation. As a result, a spherical bubble of ionized hydrogen (H II) forms around the star. It turns out that ionization progresses as ionization front, i.e. a thin spherical shell propagating outwards (see [4], Sect. 12.3). Inside the shell, virtually all hydrogen is ionized. The radial propagation of the shell is described by a differential equation for the time-dependent radius $r(t)$ [9] (convince yourself that this equation is a Bernoulli differential equation):
$$\dot{r}=\frac{1}{4 \pi r^2 n_0}\left(S_-\frac{4 \pi}{3} r^3 n_0^2 \alpha\right)$$ Here, $S_$ is the total number of ionizing photons (i.e. photons of energy greater than $13.6 \mathrm{eV}$; see Sect. 3.2.1) per unit time, $n_0$ is the number density of neutral hydrogen atoms $(\mathrm{HI})$, and $\alpha \approx 3.1 \times 10^{-13} \mathrm{~cm}^3 \mathrm{~s}^{-1}$ is the recombination coefficient. Recombination of ionized hydrogen and electrons competes with ionization of neutral hydrogen. ${ }^2$

## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|Second Order Differential Equations

In mechanics, we typically deal with second order differential equations of the form
$$\ddot{x}=f(t, x, \dot{x}),$$
where $x(t)$ is the unknown position function, $\dot{x}(t)$ the velocity, and $\ddot{x}(t)$ the acceleration of an object of mass $m$ (which is hidden as parameter in the function $f$ ). The differential equation allows us to determine $x(t)$ for a given initial position $x_0=x\left(t_0\right)$ and velocity $v_0=\dot{x}\left(t_0\right)$ at any subsequent time $t>t_0$. For this reason, it is called equation of motion.
A very simple example is free fall:
$$\ddot{x}=g,$$
where $x$ is the coordinate of the falling mass in vertical direction. Close to the surface of Earth, $g \approx 9.81 \mathrm{~m} \mathrm{~s}^{-2}$ and $f(t, x, \dot{x})=g$ is constant. Of course, you know the solution of the initial value problem for this equation:
$$x(t)=x_0+v_0\left(t-t_0\right)+\frac{1}{2} g\left(t-t_0\right)^2 .$$
For larger distances from ground, however, the approximation $f(t, x, \dot{x})=g$ is not applicable and we need to use the $1 / r$ gravitational potential of Earth. In this case,the right-hand side of the equation of motion is given by a position-dependent function $f(x)$ and finding an analytic solution is possible, but slightly more difficult. In general, $f$ can also depend on the velocity $\dot{x}$, for example, if air resistance is taken into account. We will study this case in some detail in Sect. $4.2$ and you will learn how to apply numerical methods to solve such a problem. An example, where $f$ changes explicitly with time $t$ is a rocket with a time-dependent mass $m(t)$.

To develop the tools we are going to apply in this chapter, we shall begin with another second order differential equation for which an analytic solution is known. An almost ubiquitous system in physics is the harmonic oscillator:
$$m \ddot{x}+k x=0,$$
which is equivalent to
$$\ddot{x}=f(x) \text { where } f(x)=-\frac{k}{m} x .$$

# 天体物理学和天文学代考

## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|First Order Differential Equations

$$\dot{x}=\lambda x,$$

$$\dot{x}=\alpha(t) x+\beta(t) x^\rho,$$

$$\dot{r}=\frac{1}{4 \pi r^2 n_0}\left(S_{-} \frac{4 \pi}{3} r^3 n_0^2 \alpha\right)$$

## 物理代写|天体物理学和天文学代写Astrophysics and Astronomy代考|Second Order Differential Equations

$$\ddot{x}=f(t, x, \dot{x}),$$

$$\ddot{x}=g,$$

$$x(t)=x_0+v_0\left(t-t_0\right)+\frac{1}{2} g\left(t-t_0\right)^2 .$$

$$m \ddot{x}+k x=0,$$

$$\ddot{x}=f(x) \text { where } f(x)=-\frac{k}{m} x .$$

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