This is for ##r>R## so I have to also find ##E(r<b)##, ##E(b<r<R)## ?
Then The U will be,
$$U = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{R}^{\inf}1/r^2dr + \frac{e_0}{2} \int_{b}^{R}\frac{\rho (r - b^3)}{3ε_0}dr$$
Where ##E = \frac{\rho (r - b^3)}{3ε_0}, (b<r<R)##
I tried to use ##W = ε_0/2 \int E^2d\tau## for all space. So I find that ##E = \frac{(R^3 - b^3)\rho}{3ε_0r^2}## where ##\rho## is the charge denisty. So from here when I plug the equation I get something like
$$W = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{?}^{\inf}1/r^2dr$$
Is this...
So the photon sended from ##50c/H## cannot reach the observer due to the expansion of the universe but later on when universe slows down the photon catches up the same proper distance at ##0.1c/H_0## ?
I see.
In cosmological models the relationship between proper distance to a galaxy at the emission and absorption times can be written as ##d_p(t_e)(1 + z) = d_p(t_e)##
In this case in most cosmological models we get a maximum value for the ##d_p(t_e)##. This maximum value can be also seen from...
This question will sound mostly stupid but anyways.
We see that galaxies have some velocity due to the hubble law. Lets take an object that has a recessional velocity of ##v##. In SR case assuming the universe is nearly flat, can we say that the galaxy gains mass relative to us ?
I guess in...
At this point you can assume that ##a(t) ∝ t^q## where ##q## is just a number.
So we have
¨$$\ddot{a} = q(q-1) t^{q-2}$$
$$a^{-3} = t^{-3q}$$
hence
$$q (q-1)t^{q-2} = Ct^{-3q}$$
##q - 2 = -3q ##
##q = 1/2##
so
##a(t) ∝t^{1/2}##
The OP is talking about the derivatiion of the ##1 + z = 1/a(t_e)##. In that integral we are taking ##a(t_e)## or ##a(t_0)## as constants because the time between two wave crests is too small, hence we can take then out.
In this problem we are solving for a general time.
Yes that was on my mind
distance = [0] #the initial value
for k in range(N):
d = abs(1/2 * h * (function[a + k*h] + function[a + (k+1) * h]))
distance.append(distance[-1] + d)
It should be like this.
Thansks for the tips.
I dont think energy will be conserved in the RK method. Well yes you are right, I just wanted to print them. For instance If the code runs for binary system and not for an normal orbit, we will need kinetic and potential energy values of the system. In this case avg values will not be usefull I...
Yes I have seen it.
$$\int_{-\infty}^{\infty} dx f \left(x\right) \delta \left( g \left(x \right) \right) = \Sigma f(x_i)/g'(x_i)$$ ? But I am not sure why this is the case
If the question was
$$ \int_{∞}^{∞}dxf(x)δ((x - x_1)) = ? $$ The answer would be ##f(x_1)##
So the delta function has two roots, I searched the web and some books but I am not sure what approach should I use here. I guess theres sometihng happens when ##x_1 = -x_2##.
So I am not sure what...
It passed the earth sun test. But I was more about wandering cases like binary star etc. How can assign reasonable initial conditions for such systems ?
Now it came to my mind that I can try to simulate each planet im the Solar system and investigate their energy
Yes I was careful about that
"-." lines are potential energy
"--" lines are kinetic energy and other is total energy.
My findings
Potential energy: -5.30517849127762e+33 Kinetic Energy: 2.6525789562766106e+33 Total energy: -2.6525995467214404e+33
And these values from a site...
I guess I solved it ( The question claims that we should find a non circular orbit, but near elliptical one ). Here is the graph of the orbit
Now I ll try to calculate the total energy of the system if it matches with the real values great!
I searched online but they are using acceleration... I guess Verlet algorithm is also not much accurate... I did not understand the idea .. I was trying to solve a "one" body problem by using RK method but it seemed dead end so I was trying to do it with verlet but I couldnt understand the...
I am trying to understand the verlet algorithm but I am kind of stuck.
I guess first we are findind the ##v(t + 1/2h)## then we are leaving it there and starting a loop for 8.78 ?
Also I did not understand the meaning of the equation 8.78 ? We are never using ##v(t + 3h/2)## ? Or in the...
In second order case we should rewrite the equation in terms of 2 first order DE's. So I wrote,
$$dx/dt = wx$$ $$dwx/dt = -GMx/r^3$$ and $$dy/dt = wy$$, $$dwy/dt = -GMy/r^3$$
Now I guess theres two ways to do it in 4th order RK method. I would either do it component by component or just in...
I wrote this code, but I am not sure its ture or not
from numpy import arange
from math import floor
from pylab import plot, show
Vout = 0 #inital condition
t_i = 0 #initial time
t_f = 10 #final time
N = 100000
h = (t_f - t_i) / N
RC_values = [0.01, 0.1, 1]
def f(Vout, t):
if floor(2 *...