Question

(a) The radius of the Universe is about 14.2Gpc (gigaparsecs). Convert this to meters.

(b) The density of matter in the Universe is about 3×10⁻²⁷kg m⁻³. What is the mass of the Universe?
(c) Using this mass, what is the Schwarzchild radius of the Universe, that is, the radius of a black hole with the mass of the Universe? They are not quite the same, but rather close. We have no idea if this is a coincidence or if it implies our Universe is a black hole. There is no need in cosmology that the Universe be a black hole, but ... coincidences often appear because of some underlying physical law.

Now, a couple of easy pieces, as Richard Feynman would say.
(a) Calculate the reduced mass of the Sun-Jupiter system.
(b) Calculate the distance from the center of the Sun to the barycenter of the Sun-Jupiter system (
c) Compare the barycenter location to the radius of the Sun. Is it inside or outside the Sun's surface?

Answer 1

In this question, we are given the radius of the Universe in Gigaparsecs and need to convert it to meters. We also need to calculate the mass of the Universe using the given density of matter. Furthermore, we are asked to calculate the Schwartzchild radius of the Universe and solve some easy pieces involving the Sun-Jupiter system and the barycenter location.

Step-by-step explanation:

(a) To convert Gigaparsecs to meters, we need to use the conversion factor:

1 Gigaparsec (Gpc) = 3.086 × 1025 meters

Therefore, multiplying the given value of 14.2 Gpc by the conversion factor:

14.2 Gpc × 3.086 × 1025 meters = 4.375 × 1026 meters

(b) To calculate the mass of the Universe, we need to use the formula:

Mass = Volume × Density

First, let's calculate the volume of the Universe:

Volume = (4/3) × π × (Radius)3

Volume = (4/3) × π × (4.375 × 1026)3

Next, let's substitute the given density of matter (3×10-27 kg/m3) into the formula:

Mass = Volume × Density

Mass = [(4/3) × π × (4.375 × 1026)3] × (3×10-27 kg/m3)

Calculating this expression will give us the mass of the Universe in kilograms.

(c) To calculate the Schwarzchild radius of the Universe, we can use the formula:

Schwarzchild Radius = (2 × Gravitational Constant × Mass) / (Speed of Light)2

Substitute the calculated mass of the Universe into the formula and calculate to find the Schwarzchild radius.

(a) To find the reduced mass of the Sun-Jupiter system, we can use the formula:

Reduced Mass = (Mass of Sun × Mass of Jupiter) / (Mass of Sun + Mass of Jupiter)

(b) The distance from the center of the Sun to the barycenter of the Sun-Jupiter system is called the semi-major axis of the Jupiter's orbit. This distance can be calculated using Kepler's Third Law:

Semi-major Axis = (Mass of Jupiter / (Mass of Sun + Mass of Jupiter)) × Distance between the Sun and Jupiter

(c) To compare the barycenter location to the radius of the Sun, we need to determine whether it is inside or outside the Sun's surface. The radius of the Sun is approximately 696,340 km, and the calculated semi-major axis from part (b) will give us the distance between the Sun and Jupiter's barycenter. By comparing these two distances, we can determine the relative position of the barycenter with respect to the Sun.