# Why do we use the value 206265 in the small angle formula?

Let $$d$$ be the distance, $$D$$ is the linear size, and $$x$$ the angular size.

In the formula $$D = \frac{dx}{206 265}$$

what does this number mean?

• @uhoh In general we want to encourage new users to post a reasonable amount of context, not just to help those who answer questions, but to help those who can learn from questions and answers (so they do not get confused with other things that have e.g. coincidental expressions or numbers). A little too much information is always preferred over not quite enough. :-) Jul 3 '20 at 7:54
• @uhoh Your comment here was removed, so maybe you could shed some light what happened here, please? Furthermore: Are you able to point me to which "small angle formula" is refered to here? Jun 24 at 8:25

The number 206265 arcseconds/radian is often used in astronomy for angular conversions. It is simply derived from the product of 3600 arcseconds/degree and 57.2958 degrees/radian.

### Edit based on symbols as defined in comments below

With the distance to an object, $$d$$, and its lateral dimension, $$D$$, and using the small angle approximation where $$D \ll d$$, the angular extent of the object in radians is given by $$D/d$$. Assuming that you have a measured angle, $$x$$, in arc seconds, you would first need to use the conversion factor above to find the angle in radians, resulting in the following equation.

$$\frac{x}{206265} = \frac{D}{d}$$

From that relation, knowing the measured angle and distance, you can caclulate the transverse linear dimension.

• d is the distance, D is the actual size, linear size, X is the angular size Jul 3 '20 at 3:41
• Could you write down the formula and prove it mathematically? Jul 3 '20 at 3:42
• I edited my answer based on the symbol definitions in your comment. You should edit the original question to include these definitions, as requested in several comments to the question above. Jul 3 '20 at 21:52

$$1^{c}=57.2958^{\circ}=57.3\times3600=206265″$$

$$\Longrightarrow\theta_{\rm arcsec}=\left(\frac{d}{D}\right)\times206265$$

• Welcome to astronomy SE and thanks for your contribution! Please read How to Answer for some guidance how to write good answers. In the current form, I down-voted your answer as math without any explanations is not considered good style - even if it may be correct. Jun 24 at 8:23
• @B--rian says who? Can you cite something that says so or provide evidence that "math without any explanations is not considered good style"? Where would the average user here find this wisdom? And new users don't read "style guides" before their first post, I think SE is a bit peculiar and it takes a while to get used to it.
– uhoh
Jun 25 at 4:30
• @uhoh I am just propagating what I learned at (other) SE sites, I personally love math as universal language. I retracted my downvote. Jun 25 at 6:48
• @B--rian I know what you mean, but Stack Exchange mythologies can become pervasive; anybody can say X is not considered Y here, and new users may get the impression we can make policy in the comments section. There's the help center and there's Astronomy SE Community Policy Repository (and also Space SE Community Policy Repository) I think if it's not there and maybe it should be considered, then maybe a meta question like "Can prose-less all-math answers ever be okay?"
– uhoh
Jun 25 at 7:15