How do astronomers measure an objects angular size?

[Vast]

Lets use Jupiter which has a diameter of 142,984 km at a distance from Earth of 5 AU or 747,989,353 km. The answer should be around 40.0 arcseconds, I’m just having trouble understanding how to do the calculation. If someone could help it would be much appreciated.

 

[Janus]

$$\theta = 2 \arctan{\frac{r}{d}}$$

Where $r$ is the radius of the body and $d$ is the distance to the body.

 

[jtbell]

$$\theta = 2 \arctan{\frac{r}{d}}$$

Which of course reduces to $\theta = 2r / d$ (where $\theta$ comes out in radians) in the small-angle approximation, which surely holds for Jupiter viewed from the Earth (or even the moon or sun viewed from the earth).

And then you have to convert units as necessary, e.g. between radians and arcseconds.

 

[Vast]

I’m getting about 19.114. Is that right? Which would be arcseconds, not in radians, so I shouldn’t need to convert? What am I doing wrong? Unless, I just need to multiply it by 2?

 

[SpaceTiger]

I’m getting about 19.114. Is that right? Which would be arcseconds, not in radians, so I shouldn’t need to convert? What am I doing wrong? Unless, I just need to multiply it by 2?

The number you gave is the diameter, which is already twice the radius, so all you should need to do is divide those two numbers and then convert to arcseconds:

$$\theta_{arcseconds}=\frac{diameter}{distance}\times206265$$

 

[Vast]

Thanks SpaceTiger. I got 39.429 seconds of arc, but I’m not sure where you got the number 206265 from. Can you explain that?

 

[SpaceTiger]

Thanks SpaceTiger. I got 39.429 seconds of arc, but I’m not sure where you got the number 206265 from. Can you explain that?

It's just the factor of conversion from radians to arcseconds. An arcsecond is equal to the angle subtended by earth-sun distance (AU) at a distance of parsec, so another way to write this is:

$$\theta_{arcseconds}=\frac{diameter_J}{distance_J}*\frac{parsec}{earth-sun~distance}$$

where the "J" subscripts denote quantities for Jupiter. It's just a set of units that are convenient for interstellar scales. Note that a parsec is of order the distance to the nearest star. Also note that it was the parsec that was defined from the arcsecond and AU. An arcsecond can also be converted from radians by remembering that it's just 1/60 of an arcminute, which is 1/60 of a degree, which is 1/360 of a full rotation (2*pi radians).

 

[Vast]

Ok, I now see how that number is arrived at. Thanks for the explanation.