How large should Venus appear in arc seconds? Use the small angle formula and the diameter of Venus. Note that an arc second is a 1/60th of an arc minute, and an arc minute is a 1/60th of a degree.
To determine how large Venus should appear in arc seconds, we can use the small angle formula. The small angle formula relates the apparent size of an object to its actual size and the distance to the object.
The formula is as follows:
θ = (diameter / distance) × (1 degree / 60 arc minutes) × (1 arc minute / 60 arc seconds)
In this case, we need to find the value of θ, which represents the apparent size of Venus in arc seconds. The diameter of Venus is 12,104 kilometers.
First, we need to convert the diameter of Venus to the same unit as the distance, which is usually taken as the average distance between Earth and Venus. The average distance between Earth and Venus is approximately 41 million kilometers.
Now, let's plug the values into the formula:
θ = (12,104 km / 41,000,000 km) × (1 degree / 60 arc minutes) × (1 arc minute / 60 arc seconds)
Simplifying the equation:
θ = (12,104 / 41,000,000) × (1 / 60) × (1 / 60)
θ ≈ 0.0001989 degrees
Since 1 degree is equivalent to 60 arc minutes and 1 arc minute is equivalent to 60 arc seconds, we need to convert θ to arc seconds:
θ in arc seconds ≈ 0.0001989 degrees × (60 arc minutes / 1 degree) × (60 arc seconds / 1 arc minute)
θ in arc seconds ≈ 0.0001989 × 60 × 60
θ in arc seconds ≈ 0.7160 arc seconds
Therefore, Venus should appear to be approximately 0.7160 arc seconds in size.