(a) What is the escape speed from a \(1.5 \mathrm{M}_{\odot}\) neutron star of radius \(10 \mathrm{km} ?\) (b) How does it compare with the speed of light?

The gravitational constant, denoted by the symbol \( G \), is a fundamental constant central to Newton's law of universal gravitation. It determines the strength of the gravitational force between two masses. Newton's equation is expressed as:

The constant value is \( 6.674 \times 10^{-11} \text{N·m}^2/\text{kg}^2 \), where:

• \( \text{N} \) denotes newtons, the unit of force.
• \( \text{m}^2 \) denotes square meters, representing the area.
• \( \text{kg}^2 \) denotes kilograms squared, representing the mass.

In the context of the escape speed calculation, \( G \) is used to determine the velocity required to break free from the gravitational pull of a neutron star. A higher value of \( G \) indicates a stronger gravitational force, necessitating a higher escape speed.

Understanding \( G \) is crucial as it helps quantify the gravitational influence of massive bodies like neutron stars or planets.

A neutron star is an incredibly dense remnant of a supernova explosion. These stars:

• Have a mass about 1.4 to 2 times that of the Sun (here, \( 1.5 \text{M}_{\text{☉}} \) ).
• Have a radius typically around 10 kilometers (\( 10^4 \text{m} \)).

Despite their small size, neutron stars possess a tremendous amount of mass packed into those tiny volumes, making their gravitational pull extremely strong.

The mass given in the problem, \( 1.5 \text{M}_{\text{☉}} \), must be converted to kilograms for calculations. This conversion is based on the Sun's mass, which is approximately \( 1.989 \times 10^{30} \text{kg} \). Multiplying this value by 1.5, we obtain \( M \text{neutron star} ≈ 2.9835 \times 10^{30} \text{kg} \).

Another critical property is the radius, provided as 10 km, which converted to meters is \( 10 \times 10^3 \text{m} \) or \( 1.0 \times 10^4 \text{m} \). Using these properties in the escape speed equation [[ \text{v}_e = \text{√\frac{2GM}{R}} ], ] we can calculate the incredible escape speed required to leave the neutron star’s gravitational field.

Comparing the escape speed of a neutron star to the speed of light provides insight into the immense gravitational strength of such objects.

The speed of light, denoted by \( c \), is approximately \( 3.0 \times 10^8 \text{meters per second (m/s)} \). In the problem, we calculated the escape speed (\( \text{v}_e \)): \( \text{√(3.986 \times 10^{21})} \text{m/s} \), which simplifies to approximately \( 6.31 \times 10^{10} \text{m/s} \).

Now, to gauge the magnitude of this escape speed in relation to the speed of light, we use the ratio of escape speed to light speed:

• \( \frac{v_e}{c} = \frac{6.31 \times 10^{10} \text{m/s}}{3.0 \times 10^{8} \text{m/s}} ≈ 210 \).

Notably, the escape speed is 210 times greater than the speed of light, underscoring the extreme gravitational forces present on neutron stars. Such comparisons highlight the powerful effects of gravity in these celestial objects, shedding light on their unique and fascinating nature.