One second after the Big Bang, the density of the universe...

One second after the Big Bang, the density of the universe was about $0.1 \mathrm{kg} / \mathrm{L}$. a. How big a volume of the universe contained as much mass as the Solar System? b. Estimate approximately how much bigger the universe would have been after three minutes. How big would the Solar System mass region be then? Compare this to the present size of the Solar System.

One second after the Big Bang, the density of the universe was about $0.1 \mathrm{kg} / \mathrm{L}$.
a. How big a volume of the universe contained as much mass as the Solar System?
b. Estimate approximately how much bigger the universe would have been after three minutes. How big would the Solar System mass region be then? Compare this to the present size of the Solar System.

Key Concepts

Big Bang

The Big Bang marks the origin of the universe, where all of space, time, and energy emerged from an extremely hot and dense state. Understanding the early moments of the universe is crucial for cosmology, as it sets the initial conditions for later expansion, cooling, and structure formation.

Density and Mass Relationship

Density is defined as the mass per unit volume of a substance or a region of space. In cosmology, knowing the density of the early universe allows one to calculate the volume that would contain a given mass. This relationship is essential for linking microphysical conditions to macroscopic scales in the universe.

Cosmological Expansion

Cosmological expansion refers to the process by which the universe grows in size over time. This expansion decreases the density of matter and energy in the universe and influences the scale of regions that once contained a specified mass. Understanding this concept is key to tracing the evolution of structures from the early universe to the present day.

Scale Factor

The scale factor is a parameter that quantifies the relative expansion of the universe over time. It shows how distances, volumes, and densities change as the universe evolves. The scale factor is central to models of the universe's dynamics and helps relate early-time quantities to their later-time counterparts.

Estimation Techniques in Astrophysics

In astrophysics and cosmology, estimation techniques—often involving order-of-magnitude calculations—are critical for approximating physical quantities when exact values are unknown or when dealing with very large or very small scales. These techniques enable scientists to make reasoned assessments about the properties of the universe across different epochs.

Transcript

00:01 Question number 49 is asking if the sun had a density equal to that of the universe's critical density, what would its size be? the critical density of the universe is estimated to be at a value of 10 to the minus 26 kilograms per cubic meter.

00:19 For the purposes of this problem, we're assuming that the solar mass hasn't changed.

00:26 We're going to make it so that it's still at a value of 1 .99 times 10 to the 30 kilograms.

00:37 The reason the mass is important here is because in order to find the density of an object, we divide its mass by its volume.

00:47 And the volume of a sphere is equal to a four -thirds pi times the cube of its radius.

00:53 And the radius is what we're trying to find in this problem.

00:58 We can get there by solving to find the volume.

01:02 So let's rearrange our equation for density and say that mass divided by density will be equal to volume.

01:12 So the volume of the sun will now be equal to 1 .99 times 10 to the 30 kilograms divided by 10 to the minus 26 kilograms per cubic meter.

01:28 In performing our calculation, we find that the sun's volume in this problem is now equal to 1 .99 times 10 to the 56 cubic meters.

01:43 And we're going to set that equal to 4 thirds pi r cubed.

01:48 We want to solve for r.

01:50 So, we're going to solve for r.

01:51 So let's move 4 thirds pi to the other side of the equation.

01:56 That means that r cubed will be equal to 3 fourths times 1 .99 times 10 to the 56 cubic meters over pi...

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