A neutron star is an exotic astronomical object that can result from a supernova

Question

A neutron star is an exotic astronomical object that can result from a supernova explosion (if the mass of the progenitor star is within a narrow range). A particular neutron star has a mass of 1.27 solar masses and a radius of 12.5 km. 1 solar mass = 1.98E30 kg.

A. What is the local acceleration due to gravity at the surface?

B. What is the escape velocity from the surface?

C. An electron (mass me = 9.11E-31 kg) leaves the surface with a velocity of 0.349c, where c is the speed of light (3E8 m/s). How far above the surface of the neutron star does the electron get before it comes back down?

D. Assuming the neutron star is a solid sphere with a rotational period of 41 milliseconds, what is its rotational kinetic energy?

Explanation / Answer

A. The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by 0 or n, is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2, which is exactly 35.30394 km/(h·s) (about 32.174 ft/s2, or 21.937 mph/s). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from rotation of the Earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5 percent greater at the poles than at the equator.

Although the symbol is sometimes used for standard gravity, (without a suffix) can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth (see Earth's gravity). The symbol should not be confused with G, the gravitational constant, or g, the symbol for gram. The is also used as a unit for any form of acceleration, with the value defined as above; see g-force.

The value of 0 defined above is a nominal midrange value on Earth, originally based on the acceleration of a body in free fall at sea level at a geodetic latitude of 45°. Although the actual acceleration of free fall on Earth varies according to location, the above standard figure is always used for metrological purposes. In particular, it gives the conversion factor between newton and kilogram-force, two units of force.

B. In physics, escape velocity is the minimum speed needed for an object to "break free" from the gravitational attraction of a massive body. The escape velocity from Earth is about 40,270 km/h (25,020 mph). More particularly, escape velocity is the speed at which the sum of an object's kinetic energy and its gravitational potential energy is equal to zero. If given escape velocity, the object will move away forever from the massive body, slowing forever and approaching but never quite reaching zero speed. Once escape velocity is achieved, no further impulse need be applied for it to continue in its escape. In other words, if given escape velocity, the object will move away from the other body, continually slowing and will asymptotically approach zero speed as the object's distance approaches infinity, never to return.

For a spherically symmetric massive body such as a star or planet, the escape velocity for that body, at a given distance is calculated by the formula

where G is the universal gravitational constant (G = 6.67×1011 m3 kg1 s2), M the mass of the body to be escaped, and r the distance from the center of mass of the mass M to the object.[nb 2] The relation is independent of the mass of the object escaping the mass body M. Conversely, a body that falls under the force of gravitational attraction of mass M from infinity, starting with zero velocity, will strike the mass with a velocity equal to its escape velocity.

When given a speed  greater than the escape speed  the object will asymptotically approach the hyperbolic excess speed satisfying the equation:

In these equations atmospheric friction (air drag) is not taken into account. A rocket moving out of a gravity well does not actually need to attain escape velocity to escape, but could achieve the same result (escape) at any speed with a suitable mode of propulsion and sufficient propellant to provide the accelerating force on the object to escape. Escape velocity is only required to send a ballistic object on a trajectory that will allow the object to escape the gravity well of the mass M.

D. A neutron star is a type of compact star. Neutron stars are the smallest and densest stars known to exist in the Universe.[1] With a radius of only about 11–11.5 km (7 miles), they can, however, have a mass of about twice that of the Sun. They can result from the gravitational collapse of a massive star that produces a supernova. Neutron stars are composed almost entirely of neutrons, which are subatomic particles with no net electrical charge and with slightly larger mass than protons. They are supported against further collapse by quantum degeneracy pressure due to the phenomenon described by the Pauli exclusion principle. Neutron stars are very hot and typically have a surface temperature around 6×105 K. They are so dense that a normal-sized matchbox containing neutron-star material would have a mass of approximately 5 trillion tons, or 1000 km3 of Earth rock. They have strong magnetic fields, between 108 and 1015 times as strong as that of Earth. The gravitational field at the star's surface is about 2×1011 times stronger than on Earth.

Neutron stars rotate, and can emit beams of electromagnetic radiation that are detected as pulsars. Indeed, the discovery of pulsars in 1967 first suggested that neutron stars exist. The radiation from pulsars is thought to be primarily ejected from regions near their magnetic poles. If their magnetic poles do not coincide with rotational axis of the star, it will lead to pulsations of radiation towards Earth when their magnetic poles point towards Earth during their rotation. The rotation of neutron stars can be very rapid; up to 716 times a second[6][7] has been detected, which is approximately 43,000 revolutions per minute, giving a linear speed at the surface on the order of 0.165 c.

There are thought to be around 100 million neutron stars in the Milky Way, a figure obtained by estimating the number of stars that have gone supernova.[8] However, most are old and cold, and neutron stars can only be easily detected in certain instances, such as if they are a pulsar or part of a binary system. Non-rotating and non-accreting neutron stars are virtually undetectable; however, the Hubble Space Telescope has observed one thermally radiating neutron star, called RX J185635-3754.Gamma-ray bursts may be produced from rapidly rotating high-mass stars that collapse to form a neutron star, or from the merger of binary neutron stars. Soft gamma repeaters are conjectured to be a type of neutron star known as a magnetics, or alternatively neutron stars with fossil disks around them.

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