When at closest approach to Earth, Planet X is observed to be 3.8 arcseconds in diameter. What is Planet X's diameter in meters?

Question

Christopher Ales · Accepted Answer

There is no enough data. You need to know the distance to the planet.

One can obtain an expression by: #r=l*tan(alpha/2)#, where r is the radius of the planet, l the distance to the planet, and alpha its angular width. #alpha# is a very small angle, therefore in radians: #tan(alpha)=alpha# Converting arcsecs to radians #tan(alpha)~~((alpha/s)/(3600 s/(degree)))*((pi radians)/(180 degrees))# #tan(3.8/2)~~(1.9/3600)*(pi/180)=9.2xx10^-6# Imagine for a moment that the distance is 50 million kilometers (Mars or Venus may be that far away): #r=50xx10^9*9.2xx10^-6=460xx10^3 m# (Not Mars, nor Venus.) The diameter will be 920 thousand meters.

Nova Adams · Answer

undefined To find Planet X's diameter in meters, we need to use the formula:

diameter (meters) = angular diameter (radians) × distance to planet (meters)

First, we convert the angular diameter from arcseconds to radians:

1 arcsecond = 4.848 × 10^-6 radians

Angular diameter in radians = 3.8 arcseconds × 4.848 × 10^-6 radians/arcsecond

Next, we need the distance to Planet X. Since this information is not provided, let's assume an average distance from Earth to Mars, which is about 225 million kilometers or 225,000,000,000 meters.

Diameter in meters = Angular diameter in radians × Distance to planet in meters