Assume that the 1.1 Solar mass core of a star with a zero age main sequence mass of 13 Solar masses collapses in a type II supernova event. Assume that 100% of the energy released by the collapsing core is converted into neutrinos, and that 2.5% of the neutrinos are absorbed by the overlying layers in order to power the ejection of the remnant. Here you will estimate the radius of the collapsed core remnant if sufficient energy is liberated to just barely eject the remaining non-core mass to a distance of infinity. Use values for this supergiant star from Appendix G of the text.
Answer: All stars less than 8 solar masses evolve to white dwarfs. Thus, before our 13 ZAMS star undergoes core collapse, we would expect the uncollapsed core to be electron degenerate and have a radius on the order of that of a white dwarf of similar mass. In this case, our 1.1 Solar mass core that we are interpreting as a white dwarf would have a radius around that of the radius of the Earth.
For part b) we know that the total energy to eject the material is 2.5% of the change in gravitational potential energy of the core:
\[E_{eject} = 0.025 \Delta E_c\]
If we expand the equation, we get the following:
\[\frac{GM_cM_e}{r_{avg}} = 0.025\left(\frac{GM_c^2}{r_f} - \frac{GM_c^2}{r_i}\right)\]
which, solving for \(r_f\), simplifies to
\[r_f = \frac{0.025 r_{avg} M_c r_i}{M_e r_i + 0.025 r_{avg} M_c}\]
The code chunk below estimates \(r\) to approximately 6000 km, which is 2 orders of magnitude larger than what we expect for a neutron star (~10 km).
import astropy.units as u
import astropy.constants as c
import matplotlib.pyplot as plt
import numpy as np
r_avg = 78 * u.R_sun
r_i = 1 * u.R_earth
M_star = 13 * u.M_sun
M_c = 1.1 * u.M_sun
M_e = M_star - M_c
r_f = 0.025 * r_avg * M_c * r_i / (M_e * r_i + 0.025 * r_avg * M_c)
print(r_f.to(u.km))6069.401274017802 kmImagine that the energy released during the decay of a particular radioactive species is 2.89 MeV. If 0.057 Solar masses of this species is produced in a SN event, and if this species has a mass of 56 atomic mass units and a radioactive half-life of 88.8 days, then calculate the following quantities:
Answer: The code chunk below calculates the answer to each question.
M_net = 0.057 * u.M_sun
m_amu = 56 * u.u
E_amu = 2.89 * u.MeV
half_life = 88.8 * u.d
# a)
n_atoms = M_net / m_amu
print(f"a) total number of atoms: {n_atoms.to(u.kg / u.kg):.2e} atoms")
# b)
decay_const = np.log(2) / half_life
print(f"b) decay constant: {decay_const:.2e}")
# c)
print(f"c) num decays/sec: {decay_const.to(1 / u.s):.2e}")
# d)
L = E_amu * decay_const * n_atoms
print(f"d) L (init): {L.to(u.L_sun):.2e}")
# e)
new_amount = n_atoms * np.exp(-decay_const * 1 * u.yr)
L_yr = E_amu * decay_const * new_amount
print(f"e) L (1yr): {L_yr.to(u.L_sun):.2e}")
# f)
T = np.log(100 ** (6.191/5)) / decay_const
print(f"f) Time to fade 6.191 mag: ~{T.to(u.yr):.2f}")a) total number of atoms: 1.22e+54 atoms
b) decay constant: 7.81e-03 1 / d
c) num decays/sec: 9.03e-08 1 / s
d) L (init): 1.33e+08 solLum
e) L (1yr): 7.70e+06 solLum
f) Time to fade 6.191 mag: ~2.00 yrIndividual “A” stands (miraculously and unphysically) on the surface of the more massive neutron star that gave rise the GW170817 event. Individual “B” is in a starship located 0.5 AU away from the neutron star; there is no relative motion between the starship and the neutron star.
Answer: The code chunk below calculates the values for part a) and b). Note that a radius \(r_0=10\) km was assumed as that is the typcially accepted value of a neutron star. For both parts, the following equation was used:
\[\frac{\Delta t_0}{\Delta t_\infty} = \frac{\nu_\infty}{\nu_0} = \left(1 - \frac{2GM}{r_0 c^2}\right)^{1/2}\]
where the 0 and \(\infty\) index indicate individual “A” and “B”, respectively.
Note that for part b), we can solve for \(\lambda\) by \(c=\lambda\nu\), which adds the following relation to our equation above.
\[\frac{\lambda_0}{\lambda_\infty} = \frac{\Delta t_0}{\Delta t_\infty} = \frac{\nu_\infty}{\nu_0} = \left(1 - \frac{2GM}{r_0 c^2}\right)^{1/2}\]
r0 = 10 * u.km # radius of the neutron star
M = 1.48 * u.M_sun
t_A = 1 * u.hr
# a)
t_B = t_A / (1 - 2 * c.G * M / (r0 * c.c ** 2)) ** 0.5
print(f"t_B: {t_B}")
# b)
lambda_A = 656.28 * u.nm
ratio = t_A / t_B
lambda_B = lambda_A / ratio
print(f"lambda_B: {lambda_B}")t_B: 1.3328370313320161 h
lambda_B: 874.7142869225755 nmThe Event Horizon Telescope has revolutionized our understanding of the regions surrounding supermassive black holes that reside in the dynamical centers of many massive galaxies.
On this plot, show the following physical scales with horizontal lines: 1 Earth radius, 1 Solar radius, 1 AU, and 100 AU. Clearly label each line. Plot the position of the M87 black hole on this diagram. Upload this image in the attachments section
Answer: The code chunk below calculates the Schwarzchild radius of the black hole to be around \(2.7\cdot10^4\) solar radii, and the shadow of the black hole to be around \(7.59\cdot10^4\) solar radii. We expect the shadow to be on the order of the Schwarzchild radius.
The code chunk below also creates the desired log-log plot in part b) of the Schwarzchild radius as a function of black hole mass.
M87 = 6.5E9 * u.M_sun
dist87 = 16.8 * u.Mpc
angDiam87 = 42E-6 * u.arcsec
# a)
R_s_87 = 2 * c.G * M87 / c.c ** 2
diam87 = angDiam87.to(u.rad) * dist87 / u.rad
rad87 = diam87 / 2
print(f"Schwarzchild raius: {R_s_87.to(u.R_sun):.2e}")
print(f"Radius of shadow: {rad87.to(u.R_sun):.2e}")
# b)
mass = np.linspace(1, 1E10, 1000) * u.M_sun
R_s = 2 * c.G * mass / c.c ** 2
plt.plot(mass, R_s.to(u.R_sun))
plt.axhline((1 * u.R_sun).value, linestyle="--", color="orange", label="1 $R_{\\odot}$")
plt.axhline((1 * u.au).to(u.R_sun).value, linestyle="--", color="green", label="1 AU")
plt.axhline((100 * u.au).to(u.R_sun).value, linestyle="--", color="purple", label="100 AU")
plt.plot(M87.to(u.M_sun), R_s_87.to(u.R_sun), 'o', label="M87", )
plt.legend()
plt.xlabel("Mass [$M_{\\odot}$]")
plt.ylabel("$R_S$ [$R_{\\odot}$]")
plt.xscale("log")
plt.yscale("log")Schwarzchild raius: 2.76e+04 solRad
Radius of shadow: 7.59e+04 solRad