The function vc(r) (equation (1) ) given r in [kpc] return the speed in [km/s].
Using r=8.34[kpc] of the Earth I obtain about 257 [km/s] which is not too far from 217 [km/s] that I found in wikipedia.
Computing n from equation (6) n = 30.8 [km/(s*kpc)] or about 10^(-15) [1/s] , the period of the sun using eqs. (6-8) should be
the solution of n*t = 2*pi or t = 2*pi/n about 6.3 * 10^15 [s] that converted in year is about 2 * 10^17 Myr, too much.
The unit in equations (1-3) is [kpc], the unit of equations (10-12) is [km/s] thus equation are not compatible.
It is possibile to give some extra information?
Ok, I solved by myself. Converting vc(r) in [kpc/Myr]
assuming that time unit is Myr in equations (6-12) R is in [kpc] and speed is in [kpc/Myr]
units are consistent and the period of the sun is about 200 Myr (should be correct).
However vc(r) must be multiplied by (10^6 * yr / kpc) about 0.001022712 to be consistent with the other equations.
That’s correct: you need to convert velocity to kpc/Myr (or any self-consistent set of units).
You may run into precision issues due to the large dynamic range of the numbers being used.
The last column in stars.txt can be used to check whether you’ve implemented your ephemeris correctly. In my case, for example, I can match the value to within 1E-3 (may be too high; I’m still checking all sources of rounding and truncation errors).
You can also use that value to check whether your differential equations are propagated correctly: propagate stars numerically and compare against analytical state.
Can we confirm that equations 7-8 out put is kpc and equations 10-12 output is kpc/MYr? Also, based on the description, the "t" in equations 7-12 have unit of MYr as well?
For equations (7), (8), and (9) your output will be in whatever units you use for R, nominally kpc. For (10), (11), and (12) it will be in whatever units you use for vc(R), nominally km/sec if you use expression (1) directly with r in kpc.
The units of t need to be consistent with n. If you use expression (1) directly, you will have a problem because (km/sec)/kpc will not have the right units (units need to cancel out when used as argument to a trigonometric function).
Instead, convert the output of (1) to kpc/Myr. In that manner, n = (kpc/Myr)/kpc = 1 / Myr and you can feed t in Myr to obtain the proper dimensionless units. Finally, remember to convert all angles to radians.
Long story is short: stick to kpc and Myr until and unless it demonstrates to cause numerical difficulties (related to absolute value and not units themselves).
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