What is the distance between the Sun and the Earth?

[Indranil]

Homework Statement

A new unit of length is chosen such that the speed of light in a vacuum is unity. What is the distance between the sun and the Earth in terms of the new unit, if light takes 8 mins and 20 sec to cover this distance?

Homework Equations

A new unit of length is chosen such that the speed of light in a vacuum is unity. What is the distance between the sun and the Earth in terms of the new unit, if light takes 8 mins and 20 sec to cover this distance?[/B]

The Attempt at a Solution

Sol-
The speed of light in vacuum (c) = 1 (new unit of length s–1 ) Time taken by light to reach the Earth t = 8 min + 20 s = (8 × 60 + 20) s = 500 s ∴ Distance between the sun and the Earth = Speed of light × Time x = c × t = 1 (new unit of length s–1 ) × 500 s = 500 new unit of length
*here I don't understand why C = S^-1. Could you explain, please?

 

[Charles Link]

Let $L_o$ be the distance that light travels in one second. Then $c=1.000 \, L_o/sec$. The basic units on $c$ of 'distance per unit time" do not change, and can not be changed.

 

[Orodruin]

Let $L_o$ be the distance that light travels in one second. Then $c=1.000 \, L_o/sec$. The basic units on $c$ of 'distance per unit time" do not change, and can not be changed.

This depends on the unit system. In relativity it is usual to use a unit system where $c = 1$ is dimensionless, which underlines the intricate relationship between space and time coming together as a single spacetime. In such a unit system, you can use time units to measure lengths and vice versa. Such systems are particularly popular among particle physicists. For example, the Higgs discovery paper by CMS (just as an example) does not say that the Higgs mass is 125 GeV/c^2, it just states that the Higgs mass is 125 GeV. In fact, I would say that this question is ill defined if this is not what is intended here, since it does not specify anything about which time unit is being used. The only way of having $c = 1$ is then to use the same unit for time and length, i.e., seconds. The correct argument would be that $d = c \cdot 500\, {\rm s} = 500\, {\rm s}$.

 

[haruspex]

A new unit of length is chosen such that the speed of light in a vacuum is unity.

I would say that this question is ill defined

I would go further... it is wrong.
Making the speed of light unity in the new unit requires a unit of speed to be defined, not a unit of length. If the thinking is that the distance unit - call it a lunge - equals a light-second, say, then that would make the speed of light one lunge per second, not one lunge.
Making the speed of light unity in the dimensionless sense requires more than just the invention of a new unit.

 

[Orodruin]

I would go further... it is wrong.
Making the speed of light unity requires a unit of speed to be defined, not a unit of length.

This is not true if length and time have the same dimensions. This makes velocities dimensionless and you do not need to define a unit of speed. This is how I would interpret a unit system where the speed of light is unity (i.e., identically one without dimensions).

 

[kuruman]

I think this problem redefines the meter by using $c=1$ i.s.u.^*, but explicitly keeps the second as the unit of time. Part of the problem statement is that "light takes 8 min and 20 sec to cover this distance".
@Indranil : Repeating the problem statement is not a "relevant equation".

^*i.s.u. = in some units.

 

[haruspex]

This is not true if length and time have the same dimensions.
This is how I would interpret a unit system where the speed of light is unity (

Sure, but that cannot be achieved merely by defining a new unit of distance.

 

[Indranil]

'

Let $L_o$ be the distance that light travels in one second. Then $c=1.000 \, L_o/sec$. The basic units on $c$ of 'distance per unit time" do not change, and can not be changed.

Here what is the unit of 'L₀'? meter or centimeter
Could you explain what does it really mean 'A new unit of length is chosen such that the speed of light in a vacuum is unity'?
.

 

[Orodruin]

Here what is the unit of 'L₀'? meter or centimeter

It is a length. You can measure lengths using any length unit. In fact, here it is the unit. Its value in m or cm is just a conversion factor between this new unit and m or cm. Of course it will be 100 times larger when you convert to cm. You need to separate the notions of units from the notion of physical dimension.

 

[Indranil]

It is a length. You can measure lengths using any length unit. In fact, here it is the unit. Its value in m or cm is just a conversion factor between this new unit and m or cm. Of course it will be 100 times larger when you convert to cm. You need to separate the notions of units from the notion of physical dimension.

Could you explain what does it really mean 'A new unit of length is chosen such that the speed of light in a vacuum is unity'?

 

[Chestermiller]

Isn’t the new unit of length 1 light-year, so the speed of light is 1 light-year/year?

 

[Orodruin]

Isn’t the new unit of length 1 light-year, so the speed of light is 1 light-year/year?

This is part of why the question is ambiguous. It does not specify the time unit to be used. Also, if you use a system of units where time and length does not have the same dimension then c is not unity, it is will be dimensionful and therefore subject to change in its numerical value if you change the time unit without changing the length unit.

 

[Tom.G]

A new unit of length is chosen such that the speed of light in a vacuum is unity. What is the distance between the sun and the Earth in terms of the new unit, if light takes 8 mins and 20 sec to cover this distance?

1 A.U. (Astronomical Unit)

 

[Orodruin]

1 A.U. (Astronomical Unit)

This is incorrect. You made the measured value of the distance unity instead of that of the speed of light.

 

[haruspex]

Isn’t the new unit of length 1 light-year, so the speed of light is 1 light-year/year?

Then why not a light second, making the speed of light one light second/second?
Either way, it does not make the speed of light a dimensionless 1, as the question implies.

 

[AdamSmith]

The standard answer is 1 AU. But I would like to highlight that common illustrations of Earth's orbit around sun are shown highly elliptical which is misleading, and that's what most people visualize. In reality it is almost a circle with eccentricity of 0.0167. Recently we were making simulations of solar system, and had to correct the developers on this aspect.

 

[Orodruin]

The standard answer is 1 AU

No it is not. If you put the length unit to be 1 AU your speed of light will be 1/500 AU/s.

 

[Tom.G]

Then why not a light second, making the speed of light one light second/second?

Breaking that down to details:

• A) Define speed of light to be 1 lightsecond per second. (hard to argue with? after all, the Meter was originally defined as 1/10⁷ the distance between the Earth Equator and the North Pole. just pick any arbitrary distance for a 'standard' to get units you like!)
• B) Time to travel Sun to Earth = 8Min 20Sec = 500Sec.

Ergo 1AU = 500lightseconds.

 

[haruspex]

Breaking that down to details:

• A) Define speed of light to be 1 lightsecond per second. (hard to argue with? after all, the Meter was originally defined as 1/10⁷ the distance between the Earth Equator and the North Pole. just pick any arbitrary distance for a 'standard' to get units you like!)
• B) Time to travel Sun to Earth = 8Min 20Sec = 500Sec.

Ergo 1AU = 500lightseconds.

No, you are missing my point. The question says:

A new unit of length is chosen such that the speed of light in a vacuum is unity

This is not possible.
You can

• define a new unit of speed such that the speed of light is 1 (of that unit); or
• define a new unit of length such that the speed of light is 1 of that unit per second (or whatever unit of time you care to fill in); or
• define an equivalence between time and distance dimensions such the speed of light is 1, dimensionless.

But none of those is what the question prescribes.