# Determine distance from the galactic center in DS9

I have a list of coordinates of black hole/neutron star systems in decimal degrees (ie. {(266.416,-29.009), (266.418,-29.009),...} from DS9 and would like to determine the distance from SgrA* in parsecs. To my knowledge, 25.8 parsecs ~ 1 degree, but the distance I calculate do not agree with values previously determined by a colleague.

For example, I have a list of nine very faint X-ray transients (VFXTs) and their coordinates, one of which is (266.4191667, -29.0101111). Using the distance formula, I estimated the degree distance from SgrA* - from Wikipedia, in decimal degrees, (266.4168371,-29.0078106) using the Euclidean distance formula and multiplied my value to get

(25.8 pc/deg)(0.00327404 deg) = 0.0844702 pc

from the center; but this value does not agree with any previously determined values of the VFXTs offsets.

The list I received from my colleague for VFXT offsets is

"vfxt_d = [.7209, .8605, 1.0233, 3.1395, 3.1628, 10.4651, 14.8372, 25.6744, 29.2093] "


I would appreciate some pointers to complete this project sometime today. Thanks in advance.

• 1 degree is not 25.8 parsecs at the distance to the galactic center. Start with the distance to the galactic center from our location, multiply by the tangent of the angle from the galactic center to get the distance from the center (excluding the line of site component, which may dominate) – antlersoft Mar 7 '19 at 17:14
• When you say our location do you mean the coordinates of Earth? Then I would have some value D from the Earth to SgrA* and would multiply by the angle (should I use the value X (from ( X, -Y) ) to get the distance d = D Tan[X] ? Is it fine to use the X value in degrees or should I use radians? – K F Mar 7 '19 at 21:15

You cannot estimate the distance to anything or from anything in 3-dimensional space using a 2-dimensional coordinate alone.

You may calculate an angular distance (as per Mike G's answer), then assume they are all at a common distance and translate that angular distance into a physical distance, but since they aren't all at the same distance, that won't work.

To do what you want requires an estimate of the distance to these sources.

If r is the lateral (orthogonal to line of sight) distance between the VFXT and Sgr A*, and R is the distance from the Sun to Sgr A*, and

$$\theta = \sqrt{(\Delta\alpha \cos \delta)^2 + (\Delta\delta)^2 }$$

then r = R tan θ as suggested in the comments. For the example in the question, using R = 7860 ± 140 pc [Boehle et al. 2016 via Wikipedia], I get r = 0.4216 ± 0.0075 pc, also not in your colleague's vfxt_d.

Where |θ| < 1°, tan θ ≈ θ in radians. The approximation in the question used this but was off by a factor of 5. A coefficient of 137.2 ± 2.4 pc/deg should give a better result.