How far could you see from a planet

[FactChecker]

How did you arrive to that conclusion? Surely if you walk a straight path across a random mine field you will have a higher chance to step on a mine if the mine field is 100 meters long than if it was 10.

Yes, but any section of the line of sight that is one light year long has the same probability of a star being on it as any other section that is one light year long. So the probability is just proportional to length, and sections of equal length have equal probabilities.

 

[StrangeCoin]

Yes, but any section of the line of sight that is one light year long has the same probability of a star being on it as any other section that is one light year long. So the probability is just proportional to length, and sections of equal length have equal probabilities.

Probability is inversely proportional to length, yes, and it is also inversely proportional to the size and number of the stars per unit volume of space.

However, I don't think we can get probability from purely geometrical relations, I believe we need to wrap that up into some probability equation.

The other thing is that the equation suggested as the answer deals with solid angles and field of view, some sort of perspective projections, but we see in the first sentence above those things are not really a part of the equation.

 

[mfb]

10*pi*(radius of sun)^2

Is this circle area multiplied by 10 stars in the first shell? How can this describe density if the number of stars and their total coverage area is not related per some defined field of view area?

Take the whole field of view all around you. A sphere with a radius of 10 light years has an area of 4 pi (10 light years)^2

field of view = 30 degrees
1st shell number of stars = 10

Now you modify the star density - but that does not change the general idea, you just get some other prefactor.

OUTPUT:
number of stars visible in the 2nd shell = ??

The expected number of visible stars is the total number of stars multiplied by the fraction of empty space in the first shell. If you don't choose an arbitrary 30 degrees field but take the full shell, the fraction occupied by the first shell is:

(10*pi*(radius of sun)^2 / (4 pi (10 light years)^2)

That's such a simple concept, I'm running out of ideas how to explain this in even more different ways.

 

[FactChecker]

The other thing is that the equation suggested as the answer deals with solid angles and field of view, some sort of perspective projections, but we see in the first sentence above those things are not really a part of the equation.

As the distance increases, the arc blocked by one star decreases, but also the arc not blocked decreases by exactly the same factor. Since the stars are infinite and remain spaced on average at 10 light years, there are always enough stars to keep the percentage of blocked sky at a fixed average no matter how far away the "shell" being calculated is. So the odds of a particular line of sight being blocked is the same in the hundredth light year distance as it is in the first light year distance.

 

[StrangeCoin]

Take the whole field of view all around you. A sphere with a radius of 10 light years has an area of 4 pi (10 light years)^2

All the terms in the equation are properties of the 1st shell, and there is no any information about the 2nd shell?

So visibility of the 2nd shell does not depend on its distance from the 1st shell, nor it depends on the distance to the observer?

 

[mfb]

All the terms in the equation are properties of the 1st shell, and there is no any information about the 2nd shell?

The thing you quoted is for the first shell, but the second shell just gets an additional factor of 4 both in the numerator and denominator, and those cancel.

So visibility of the 2nd shell does not depend on its distance from the 1st shell, nor it depends on the distance to the observer?

The shells are chosen to be equidistant in this approach, otherwise it is messy to implement a constant star density. The visible fraction of the second shell (= the fraction of shell 1 that is not blocked!) does not depend on the distance of the shell, right.