Calculating apparent separation

[bobo1455]

I am trying to calculate the apparent separation between the 2 stars Mizar and Alcor. In the question I am given the total distance from me to the 2 stars which is 83 light years and the angle of separation which is 12 arcminutes.

The question is multiple choice: (A) 0.1 ly, (B) 0.3 ly, (C) 1 ly, (D) 3 ly, (e) 10 ly

What I have tried is using the formula from Wikipedia on Angular Distance to get a value that matches one from the choices A to E

\theta ≈ \frac {a}{D}

The problem is, I am having a hard time understanding what to exactly do with 83 light years and 12 arcminutes. I have read online that the 2 stars Mizar and Alcor have distances of 81 light years and 78 light years, so the distance between them is 3 light years, so I'm assuming the answer is (D), but I don't know at all how to get there.

I am also given these formulas:

distance = actual size / angular size

angular size is proportional to actual size / distance

Any help is appreciated.

 

[BvU]

The angle of separation is the angle between the lines from you to the stars. Theta in the formula.
Don't forget to convert to radians.

 

[bobo1455]

Okay, I convert 12 arcminutes to degrees (0.2) to radians (0.0034906585) and plugged in for theta:

then I have:

0.0034906585 = x / 83 ly (not sure if I am supposed to convert ly to AU or some other unit of measurement)

then I get x = 0.2897246555, which is approx 0.3 ly, and (B) is 0.3 ly, but I am still not 100% sure it is correct or not.

 

[HallsofIvy]

The original distances were given in light years so the answer should be given in light years. Yes, 12 minutes is 12/60= 1/5 degrees and so (1/5)(3.14159/180)= 0.0034906585 so the distance between them, along a circular arc, is (0.0034906585)(83)= 0.2897 light years. I don't think I would put it to as many significant figures as you do. Since "12 arcminutes" and "83 light years" both have two significant figures, the most reasonable answer is 0.29 light years.

You could also do this as a "trig" problem. Dropping a vertical from the vertex (the earth) to the line between the two stars, we have two congruent right triangles with angle 6 arcminutes and hypotenuse 83 ly. The opposite side of that right triangle (half the distance between the two stars), x, satisfies $$sin(6')= x/83$$ so that x= 83sin(6')=0.1449 so that the entire distance, 2x, is 0.2898 light years, still 0.29 ly as before.

 

[HalfLostAndSearching]

As a scientist, let me first clarify the terms being used in this question. Apparent separation refers to the angular distance between two objects as seen from a specific point of observation, in this case, from the Earth. It is different from the actual physical distance between the two stars, which is given as 83 light years in this question. The angle of separation, or angular size, is the angle between the two stars as seen from Earth, which is given as 12 arcminutes.

To calculate the apparent separation between Mizar and Alcor, we can use the formula for angular distance, which is θ ≈ a/D, where θ is the angular size, a is the actual size, and D is the distance. In this case, we are given the distance (D = 83 ly) and the angular size (θ = 12 arcminutes). However, we do not have the actual size of the stars, so we cannot use this formula directly.

Instead, we can use the formula for angular size, which is proportional to actual size / distance. This means that the angular size of an object is directly proportional to its actual size and inversely proportional to its distance. So, we can set up a proportion using the given values and solve for the actual size of the stars.

12 arcminutes = actual size / 83 ly

Solving for the actual size, we get:

actual size = 12 arcminutes * 83 ly = 996 arcminutes * ly

Now, we can use this actual size value in the formula for angular distance to calculate the apparent separation between Mizar and Alcor:

θ ≈ a/D = 996 arcminutes * ly / 83 ly = 12 arcminutes

Therefore, the apparent separation between Mizar and Alcor is 12 arcminutes, which is also the given value in the question. This means that the answer is (C) 1 ly, as this is the only option that matches the calculated apparent separation.

In conclusion, to calculate the apparent separation between two stars, we need to use the formula for angular size and set up a proportion to solve for the actual size, which can then be used in the formula for angular distance. It is also important to understand the difference between apparent separation and actual distance, as they are two different concepts in astronomy.