Question

It is conjectured that a "burned-out" star could collapse to a "gravitational radius," defined as the radius for which the work needed to remove an object of mass \(m\) from the star's surface to infinity equals the rest energy \(m c^{2}\) of the object. Show that the gravitational radius of the Sun is \(G M_{\mathrm{S}} / c^{2}\) and determine its value in terms of the Sun's present radius. (For a review of this phenomenon see "Black Holes: New Horizons in Gravitational Theory," by Philip C. Peters, American Scientist, September- October \(1974, \mathrm{p} .575 .)\)

Short answer

The gravitational radius of the Sun in terms of the Sun's present radius is given by \(r_{S} / R_{S} = G M_{S} / (c^2 R_{S})\), where \(G\) is the gravitational constant, \(M_{S}\) is the Sun's mass, \(R_{S}\) is the Sun's present radius, and \(c\) is the speed of light.

Step-by-step solution

Set up the work-energy equation

In this scenario, the work done to move the object from the star's surface to infinity should be equal to the rest energy of the object. Therefore, we start with the work-energy equation: \[ GMm / r = mc^2 \] where \(G\) is the gravitational constant, \(M\) is the mass of the star, \(m\) is the mass of the object, \(r\) is the distance from the center of mass (radius of the star in this case), and \(c\) is the speed of light.

Solve for the radius \(r\)

Solving the above equation for \(r\) gives us: \[ r = GM / c^2 \] which represents the gravitational radius (the radius where the work done to remove an object to infinity is equal to its rest energy).

Apply values for the Sun

Substitute \(M\) with \(M_{S}\), the mass of the Sun, to find the gravitational radius of the Sun: \[ r_{S} = G M_{S} / c^2 \] To determine the gravitational radius of the Sun in terms of the Sun's present radius \(R_{S}\), divide both sides by \(R_{S}\). This will give us: \[r_{S} / R_{S} = G M_{S} / (c^2 R_{S})\] However, the gravitational constant \(G\) and the speed of light \(c\) have constant values that can be found in any physics textbook and the mass and radius of the Sun are well-determined astronomical quantities. So, you can substitute in the necessary values to find a numerical result.

Key concepts

Black Holes

Black holes are one of the most fascinating and enigmatic objects in the universe. Born from the remnants of massive stars that have exhausted their nuclear fuel, these astrophysical phenomena represent the ultimate victory of gravity over all other forces.

At the end of its life cycle, a star with sufficient mass will collapse under its own gravity to an incredibly small point known as a singularity. Surrounding the singularity is the event horizon, the so-called point of no return. The gravitational pull at the event horizon is so strong that not even light, the fastest thing in the universe, can escape from it.

The intriguing notion of the gravitational radius is closely related to black holes, where it defines the critical radius at which an object's gravitational energy equals its rest mass energy. This concept is particularly important because it establishes a fundamental limit based on the energy principles that govern these dark enigmas of space.

Work-Energy Equation

The work-energy equation is a fundamental concept in physics that describes the relationship between the work done on an object and the change in its energy. In its most basic form, the equation states that the work done by all forces acting on an object is equal to the change in its kinetic energy.

However, in the context of gravitational fields, such as the case with black holes, the work-energy equation takes a more specific form. Here, the equation enables us to calculate the work required to move an object from the gravitational field of a celestial body, like a star or planet, to a point at infinity where the gravitational influence is negligible. As the step-by-step solution illustrates, by equating this work to the rest mass energy of the object using Einstein's famous equation (\(E=mc^2\)), we can derive critical radii like the gravitational radius of the Sun.

Astrophysical Phenomena

The universe is home to a vast array of astrophysical phenomena, each showcasing the rich tapestry of physical laws that govern celestial objects and their interactions. From the birth of stars in nebulae to the violent deaths leading to supernovae and black holes, astrophysics provides a window into the most extreme environments and events in the cosmos.

Astrophysical phenomena like supernovae, pulsars, and black holes are particularly interesting in the context of gravitational theory. They exhibit strong gravitational effects that cannot be found on Earth and challenge our understanding of physics. Exploring phenomena like gravitational radius helps astrophysicists to decipher the extreme physics that rule over these mysterious corners of the universe and contribute to our overall understanding of gravity's role in cosmic evolution.

Gravitational Theory

Gravitational theory is a cornerstone of modern physics, describing how matter interacts with the curvature of spacetime itself to create what we perceive as the force of gravity. Isaac Newton first described gravity as a force that acts instantaneously across distance, but it was Albert Einstein's theory of General Relativity that depicted gravity as the warping of spacetime around massive objects.

Gravitational theory not only explains the motion of planets and the formation of tides but also enables us to understand phenomena on a much grander scale, such as the orbits of galaxies and the expansion of the universe. The concept of gravitational radius directly stems from gravitational theory, encapsulating the idea that there is a radius where the spacetime curvature created by an object's mass becomes so intense that it defines a boundary between what can escape and what is forever trapped by the object's gravitational pull.