# HW 5

## Imports


```python
import numpy
import sympy
from astropy import units, constants
```

## Theory A

### Quadratic Geometric Quantities


```python
alpha, beta = sympy.symbols('alpha beta')
```


```python
def g(mu: sympy.Symbol, nu: sympy.Symbol):
    return sympy.Symbol('g_{}_{}'.format(mu.name, nu.name))

def christoffel(mu, nu, lam):
    sympy.Derivative()
```


```python

```


```python

```


```python

```

## Problem C

### C1

Radius of orbit of binary: (7.144)

$$\omega = \bigg(\frac{GM}{4R^3}\bigg)^2 \implies R(\omega) = \left(\frac{GM}{4\omega^2}\right)^{1/3}$$


```python
def rad(omega, mass):
    return (constants.G * mass / (4 * omega ** 2)) ** (1/3)
```


```python
M1 = 36 * constants.M_sun
M2 = 29 * constants.M_sun
M_avg = (M1 + M2) / 2.0

omega = 350 * (1 / units.second) #* (2 * numpy.pi)
```


```python
r = rad(omega, M_avg)
r
```




$206474.44 \; \mathrm{m}$




```python
R_s = 2 * constants.G * M_avg / constants.c ** 2
R_s
```




$95980.627 \; \mathrm{m}$




```python
r / R_s
```




$2.1512096 \; \mathrm{}$



### C2

Distance

$$\LARGE\bar{h}_{ij} = \frac{2G}{r} \ddot{Q}_{ij}(t - r)$$

$$\LARGE \ddot{Q}_{ij} = 2Mr^2\omega^2 \begin{vmatrix}-\cos{2\omega(t-d)} & -\sin{2\omega(t-d)} & 0 \\ -\sin{2\omega(t-d)} & \cos{2\omega(t-d)} & 0 \\ 0 & 0 & 0 \end{vmatrix}$$

$$\LARGE\bar{h}_{ij} =  \frac{4GMr^2 \omega^2}{d} \begin{vmatrix}-\cos{2\omega(t-d)} & -\sin{2\omega(t-d)} & 0 \\ -\sin{2\omega(t-d)} & \cos{2\omega(t-d)} & 0 \\ 0 & 0 & 0 \end{vmatrix}$$

$$\max \bar{h}_{ij} = 4GMr^2 \omega^2 / d$$


```python
d = (410 * units.Mpc).to(units.meter)
omega_max = 350 * (1 / units.second) #* (2 * numpy.pi)
r_min = rad(omega_max, M_avg)
```


```python
h_ij_max = 4 * constants.G * M_avg * (r_min ** 2) * (omega_max ** 2) / (d * (constants.c ** 4))
h_ij_max
```




$8.8167083 \times 10^{-22} \; \mathrm{}$



### C3

$$P \sim G \dddot{Q}^2 \sim P_o \frac{r_s}{r}$$


```python
constants.G
```




$6.6743 \times 10^{-11} \; \mathrm{\frac{m^{3}}{kg\,s^{2}}}$




```python
P_o = (constants.c ** 5 / constants.G).to(units.Watt)
P_o
```




$3.6282549 \times 10^{52} \; \mathrm{W}$




```python
constants.L_sun
```




$3.828 \times 10^{26} \; \mathrm{W}$




```python
P_o / constants.L_sun
```




$9.4781999 \times 10^{25} \; \mathrm{}$




```python
P_sun_mass = (constants.M_sun * (constants.c ** 2) / (1 * units.second)).to(units.Watt)
P_sun_mass
```




$1.7870937 \times 10^{47} \; \mathrm{W}$




```python
P_o / P_sun_mass
```




$203025.45 \; \mathrm{}$



### C4

$$P(r) =  \frac{2}{5c^5} \frac{G^4 M^5}{R^5}$$

$$R(\omega) = \left(\frac{GM}{4\omega^2}\right)^{1/3}$$

$$\LARGE P(\omega) =  \frac{2}{5c^5} \frac{1}{G} \left(2GM\right)^{10/3} \omega^{10/3}$$

$$\LARGE E \equiv \int_{0}^{t_f} P(\omega(t)) dt =  \frac{2}{5c^5} \frac{1}{G} \left(2GM\right)^{10/3} \int_{0}^{t_f} \omega^{10/3}(t) dt$$

Assumption:

$$\LARGE \omega(t) = \frac{t}{t_f}\omega_f + \frac{(t_f - t)}{t_f}\omega_o = \frac{\omega_f - \omega_o}{t_f} t + \omega_o$$

$$\LARGE E = \frac{2}{5c^5} \frac{1}{G} \left(2GM\right)^{10/3} \frac{3}{13} \frac{t_f}{\omega_f - \omega_o} (\omega_f^{13/3} - \omega_o^{13/3})$$


```python
omega_init = 35 * (1 / units.second) #* (2 * numpy.pi)
omega_fin = 350 * (1 / units.second) #* (2 * numpy.pi)
```


```python
P_units = constants.G ** 4 * units.kg ** 5 / (units.m ** 5) / (constants.c ** 5)
P_units
```




$8.1944419 \times 10^{-84} \; \mathrm{\frac{m^{2}\,kg}{s^{3}}}$




```python
P_units = (1 / constants.c ** 5) * (1 / constants.G) * (constants.G * units.kg) ** (10/3) * (1 / units.second) ** (10/3)
P_units
```




$7.4617926 \times 10^{-67} \; \mathrm{\frac{m^{2}\,kg}{s^{3}}}$




```python
delta_t = 0.2 * units.second
```


```python
E = ( (2 / (5 * constants.c ** 5))
     * (1 / constants.G) * (2 * constants.G * M_avg) ** (10/3) 
     * (omega_fin ** (13/3) - omega_init ** (13/3)) * (delta_t / (omega_fin - omega_init)) * (3 / 13)
)
E
```




$5.0482335 \times 10^{47} \; \mathrm{\frac{m^{2}\,kg}{s^{2}}}$




```python
E / (constants.M_sun * constants.c ** 2)
```




$2.8248287 \; \mathrm{}$




```python

```


```python

```

8ec4063d1d21001b915556c0494c830ddd7324d1
ipynb
Jupyter Notebook
notebooks/homework-5.ipynb
JWKennington/CourseGR1
ba81a1bf0427210c14d4e97c8f2cf8a345297812
MIT
notebooks/homework-5.ipynb
JWKennington/CourseGR1
ba81a1bf0427210c14d4e97c8f2cf8a345297812
MIT
notebooks/homework-5.ipynb
JWKennington/CourseGR1
ba81a1bf0427210c14d4e97c8f2cf8a345297812
MIT
Qwen/Qwen-72B
1. YES
2. YES