Understanding Big R(t) in Astronomy: A Physical Perspective

[scoopaloop]

I'm having trouble with with understanding what this is, our text, Astronomy: a physical Perspective by Kutner, uses R(t). I understand r(t) is the distance between two objects at a point in time, but what is the scale factor big R(t)?

 

[cristo]

I'm having trouble with with understanding what this is, our text, Astronomy: a physical Perspective by Kutner, uses R(t). I understand r(t) is the distance between two objects at a point in time, but what is the scale factor big R(t)?

The scale factor describes the expansion of the universe. So, if the distance between two points at some initial time is d_0, after a time t, the distance between the points will be d(t)=R(t)d_0

 

[Chalnoth]

The scale factor describes the expansion of the universe. So, if the distance between two points at some initial time is d_0, after a time t, the distance between the points will be d(t)=R(t)d_0

Not quite. This assumes that $R(t)=1$ at the same time as $d_0$. The more correct way of stating it is:

$${d(t_1) \over R(t_1)} = {d(t_2) \over R(t_2)}$$

For me, it's a bit easier to understand if you use the $a(t)$ notation, where the current scale factor is defined to be $a(t=now) = 1$. This simplifies things to be more like your equation above:

$$d(t) = a(t) d_0$$

Where $d_0$ is defined as the current distance.

 

[scoopaloop]

I'm not sure I get it, could you explain it in the sense of the book I am using, on the right side of the page.

 

[Chalnoth]

I'm not sure I get it, could you explain it in the sense of the book I am using, on the right side of the page.

View attachment 81452

Looks like they're using the form usually attributed to $a(t)$. Notation can be confusing sometimes :P

The basic jist of it is: scale factor doubles, distances between objects doubles. So if two galaxies were a billion light years apart when the scale factor was 0.5, then those two galaxies are currently two billion light years apart (currently the scale factor is defined as 1, using their notation).

 

[scoopaloop]

Okay, I think I get it. It just gets confusing to me when they throw so many different r(t)s and rs in there. Thanks.

 

[cristo]

For me, it's a bit easier to understand if you use the a(t)a(t) notation, where the current scale factor is defined to be a(t=now)=1a(t=now) = 1. This simplifies things to be more like your equation above:

d(t)=a(t)d0​

d(t) = a(t) d_0

Where d0d_0 is defined as the current distance.

Looks like they're using the form usually attributed to a(t)a(t). Notation can be confusing sometimes :P

I thought R(t) and a(t) were just different notation for the same thing (I've never seen the version of R(t) that you define above, before). You're right, notation can be confusing!

 

[Chalnoth]

I thought R(t) and a(t) were just different notation for the same thing (I've never seen the version of R(t) that you define above, before). You're right, notation can be confusing!

That's what makes it a bit confusing. Usually $a(t)$ is defined so that the current scale factor is equal to one, and is considered a unitless parameter. Usually $R(t)$ is defined so that $k = {1, 0, -1}$. This makes it so that $R(t)$ takes units of length and can be interpreted as a radius of curvature.