Astronomy - Distance with different scale factors

• tosv
In summary, the conversation discusses the expansion of the universe and the scale factor used to describe it. It also mentions the distance between two points in a flat universe and how it changes over time. The solution involves finding the constants C_{1} and C_{2} and considering the use of different units for convenience.
tosv
Homework Statement
Expansion of the universe is described by the scale factor R(t), where t is the time after the Big Bang. For a flat universe the scale factor is today
$R(t)=C_{1}\cdot t^{\frac{2}{3}}$

When the Universe was radiation dominated, for t <200,000 years, the scale factor was
$R(t)=C_{2}\cdot \sqrt{t}$

Today, about 12 billion years after the Big Bang, we measure the distance to another superhop to 1 billion light years. How big would the distance between these sites one year after the Big Bang, if we assume that we live in a flat universe.

The attempt at a solution
I express the distance in meters
$R(t)=9.461\cdot 10^{24}\, m$

Then I express the time in seconds
$t=3.787\cdot 10^{17}\, s$

I found that the constant is
$C_{1}=1.8075\cdot 10^{13}$

I need to know the constant $C_{2}$, but I'm not sure how I'll be able to calculate it.

I have tested just to put $C_{1}=C_{2}$, but the result for the distance was wrong. I then realized that the two constants have different units.

Does anyone have a suggestion how I may proceed?

For t = 200,000 years, both expressions for R(t) should give the same result.

To make your life easier, consider using units of years and light-years rather than seconds and meters.

Thanks for your reply, now I was able to solve the problem.

I know we get SI units drilled into our heads in intro physics classes, but they are not always the most convenient. If the problem statement uses alternate units, that's a hint for you to consider it too.

I would first check the units of the two constants to make sure they are consistent. C1 has units of meters per second, while C2 has units of meters per second squared. This means that they cannot be equal to each other, as they represent different physical quantities.

To find C2, we can use the given information that the scale factor for a radiation-dominated universe (t < 200,000 years) is R(t)=C2√t. We know that the scale factor at t=3.787x10^17 seconds is 9.461x10^24 meters. Plugging this into the equation, we get:

9.461x10^24 = C2√(3.787x10^17)

Solving for C2, we get C2=6.21x10^6 meters per second. This means that the distance between two sites one year after the Big Bang would be:

R(1 year)=6.21x10^6√(3.1536x10^16)

=1.425x10^20 meters

This is a very large distance, which is expected given the rapid expansion of the universe during its early stages. However, it is important to note that this calculation assumes a flat universe, which is still a topic of ongoing research and debate in the field of astronomy. Further studies and observations may provide more accurate values for the scale factors and distances in the early universe.

1. What is the difference between astronomical units and light years?

Astronomical units (AU) and light years (ly) are both units of measurement used in astronomy to describe distances. An AU is the average distance between the Earth and the Sun, which is approximately 149.6 million kilometers. A light year, on the other hand, is the distance that light travels in one year, which is equivalent to about 9.46 trillion kilometers.

2. How does the scale factor affect astronomical distances?

The scale factor is a numerical value used to represent the size of an object or distance in relation to its actual size. In astronomy, the scale factor is often used to represent the distance between celestial bodies. For example, a scale factor of 1:100 means that 1 unit on a map represents 100 units in real life. This can be applied to astronomical distances, where a scale factor of 1:1000 would represent a much larger distance compared to a scale factor of 1:100.

3. Can astronomical distances be measured in kilometers or miles?

Yes, astronomical distances can be measured in kilometers or miles, but these units are often not practical due to the vast distances involved in astronomy. Kilometers and miles are more commonly used to measure distances within our solar system, while astronomical units and light years are used for larger distances in space.

4. How do scientists measure distances in space?

Scientists use a variety of methods to measure distances in space, depending on the scale of the distance and the object being observed. For closer objects within our solar system, they may use radar or parallax measurements. For more distant objects, techniques such as redshift and standard candles are used to calculate distances.

5. How does the distance between objects in space affect their interactions?

The distance between objects in space can greatly affect their interactions. Objects that are closer together may have stronger gravitational or electromagnetic interactions, while objects that are farther apart may have weaker interactions. This can impact the behavior and movement of celestial bodies, and plays a crucial role in understanding the dynamics of our universe.

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