i) ii) Let's consider a neutron star with radius 10 km and a density of 4 x 10^4 g/cm (4 x 10'? kg/m²). What is the esca

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I Ii Let S Consider A Neutron Star With Radius 10 Km And A Density Of 4 X 10 4 G Cm 4 X 10 Kg M What Is The Esca 1 (106.08 KiB) Viewed 80 times

I Ii Let S Consider A Neutron Star With Radius 10 Km And A Density Of 4 X 10 4 G Cm 4 X 10 Kg M What Is The Esca 2 (44.35 KiB) Viewed 80 times

i) ii) Let's consider a neutron star with radius 10 km and a density of 4 x 10^4 g/cm (4 x 10'? kg/m²). What is the escape velocity at the surface of the neutron star in km/s? Imagine that someone somehow managed to survive on the surface of the neutron star and flashed a light towards an observer far away from the neutron star. As the photons escape the gravitational potential of the neutron star, they must gain potential energy. However, light travels at a constant velocity c, and cannot lose kinetic energy to conserve energy. Instead, the photons increase in wavelength to conserve energy (the energy of a photon E = hc/). The distant observer would therefore see the photons arriving at a longer wavelength than the person flashing the light on the neutron star surface. This is gravitational redshift and has some analogy to cosmological redshift that we have discussed before. The change in wavelength of the photons can be written as: A = (1-P) too de = 1 rs Re iii) where 7, is the wavelength of the light as measured by the observer at infinity and le is the wavelength measured at the source of emission, r, is the Schwarzschild radius for a black hole with the mass of the neutron star and R, is the radius at which the photon was emitted, i.e. the radius of the neutron star. Rewrite the right side of this equation to just include the speed of light, c, and the escape velocity at the neutron star surface, ve. If the person on the surface of the neutron star uses a sodium bulb that produces photons with wavelength 589 nm, what is the wavelength of the photons seen by the observer at a large distance from the neutron star? Given the change in wavelength of the photons, can you make any educated guess as to whether time is elapsing at the same rate on the neutron star as for the observer? If it isn't passing at the same rate, how does the passing of time compare between the two frames? Clue: perhaps recall the movie Interstellar (if you saw it!), which demonstrated this effect. iv)

v) By what factor would you have to increase the density (assuming the same radius) to turn this neutron star into a black hole? What is the Schwarzschild radius of this black hole? From far away, we should not see light escape from the Schwarzschild radius as the escape velocity is equal to the speed of light at this location. This must mean that light is stopped at the Schwarzschild radius from the point of view of an external observer. However, if an observer is located at the Schwarzschild radius, they must see light travel at velocity c, as required by general relativity. Can you comment (in 1 to two sentences) on what this must imply for time at the location of the Schwarzschild radius?