Calculating Weight on a Neutron Star: A Fun Physics Problem

Imagine a scenario where you weigh 665 Newtons (N) on Earth. How much would your weight be on the surface of a hypothetical neutron star that has the same mass as the Sun but with a diameter of 17.0 kilometers? This is a fascinating problem that combines our understanding of gravity and the properties of neutron stars. Let's explore the solution step-by-step.

Understanding the Relationship Between Gravity and Distance

The force of gravity between two masses can be described by Newton's law of universal gravitation, given by the equation:

F G * m1 * m2 / r^2

Where:

F is the force of gravity (in Newtons, N) G is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2) m1 and m2 are the masses of the two objects (in kilograms, kg) r is the distance between the centers of the two objects (in meters, m)

In our scenario, the masses of the two objects (you and the neutron star) remain constant, so the only variable we need to consider is the radius (r) of the neutron star. The diameter of the neutron star is given as 17.0 kilometers, so the radius is half of that, which is 8.5 kilometers.

Comparing Gravity on the Sun and on the Neutron Star

To find the acceleration due to gravity on the neutron star in units of Earth's gravity, we need to compare it to the acceleration due to gravity on the Sun. The acceleration due to gravity on the Sun's surface can be calculated using the formula:

a G * Msun / Rsun^2

Where:

Msun is the mass of the Sun (approximately 1.989 x 10^30 kg) Rsun is the radius of the Sun (approximately 696,340 kilometers, or 6.9634 x 10^8 meters)

Given that the diameter of the neutron star is 17.0 kilometers, its radius is 8.5 kilometers, or 8.5 x 10^3 meters. Now, let's calculate the ratio of the radius of the Sun to the radius of the neutron star:

Rsun / Rneutron star (6.9634 x 10^8 meters) / (8.5 x 10^3 meters) ≈ 82,000

Squaring this ratio gives us the factor by which the acceleration due to gravity on the Sun's surface will be increased on the neutron star:

(82,000)^2 ≈ 6.72 x 10^9

To find the acceleration due to gravity on the neutron star in units of the Sun's gravity, we multiply this factor by the surface gravity of the Sun:

6.72 x 10^9 * 28 ≈ 1.88 x 10^11

Now, to convert this acceleration into units of Earth's gravity (g), we note that the acceleration due to gravity on Earth is approximately 9.8 m/s^2, or 1 g. Therefore, the acceleration due to gravity on the neutron star is:

1.88 x 10^11 * 1 g ≈ 1.88 x 10^11 g

Converting this back to weight, we multiply by the weight on Earth:

665 N * 1.88 x 10^11 g ≈ 1.25 x 10^14 N

Conclusion

So, if you weigh 665 N on Earth and you were on the surface of a neutron star with the same mass as the Sun but a diameter of 17.0 kilometers, your weight would be approximately 1.25 x 10^14 N. This is an incredibly large number, highlighting the immense gravitational forces present on neutron stars.

Remember, the solution to this problem required understanding the fundamental principles of gravity and applying them to a highly compact object like a neutron star. To fully explore this topic, you can look up the radius of the Sun and perform the arithmetic yourself. This exercise might not solve your homework for you, but it will deepen your understanding of astrophysics and the incredible forces at play in the universe.