Inconsistency between Horizons Cartesian Vector and Elements Ephemerides

Question

I am puzzled why the 'Argument of Perifocus' in the 'Elements' ephemeris of JPL Horizons data seems to change at almost twice the rate compared to that derived from the 'Cartesian Vector' data (by looking up the point in time where the radial velocity changes the sign and getting the angle by arctan(y/x) for the corresponding coordinates x,y). For Mercury, the result from the latter method is consistent with the published value for its precession, but for the Elements ephemeris it is almost a factor 2 too high. Am I missing something as far as the definition of the 'Argument of Perifocus' is concerned? Below the figures from the Horizons output for the two cases:

Target body name: Mercury (199)                   {source: DE431mx}
Center body name: Sun (10)                        {source: DE431mx}
Center-site name: BODY CENTER
Output units    : AU-D                                                         
**Output type     : GEOMETRIC cartesian states**
Output format   : 3 (position, velocity, LT, range, range-rate)
Reference frame : ICRF/J2000.0                                                 
Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch  


 260449.687500000 = **B.C. 4000-Jan-27 04:30:00.0000** TDB 
 **X = 1.164743462150131E-01 Y = 2.850041983620870E-01** Z = 8.389268356192319E-03
 VX=-3.128451713699121E-02 VY= 1.265881361215312E-02 VZ= 4.254899234299577E-03
 LT= 1.778860079563158E-03 RG= 3.080000750548330E-01 RR=-1.081697566119120E-06

**w1=arctan(y/x)=67.771 deg**


3182087.729166667 = **A.D. 4000-Feb-28 05:30:00.0000** TDB 
 **X = 5.116306925306278E-02 Y = 3.023233324889192E-01** Z = 2.103345767836515E-02
 VX=-3.347343877524768E-02 VY= 5.427345568155732E-03 VZ= 3.354722757033892E-03
 LT= 1.775062722517847E-03 RG= 3.073425830640897E-01 RR=-3.999603040263569E-06

**w2=arctan(y/x)=80.395 deg**

**w2-w1 = 12.62 deg/8000.08 years = 568 arcsec/100 year**s (published value 574 arcsec/100 years)


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Target body name: Mercury (199)                   {source: DE431mx}
Center body name: Sun (10)                        {source: DE431mx}
Center-site name: BODY CENTER
Output units    : AU-D, deg, Julian Day Number (Tp)                            
**Output type     : GEOMETRIC osculating elements**
Reference frame : ICRF/J2000.0                                                 
Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch 


 260449.687500000 = **B.C. 4000-Jan-27 04:30:00.0000** TDB 
 EC= 2.043363097473196E-01 QR= 3.080000741369763E-01 IN= 7.355089152666030E+00
 OM= 5.558492230906509E+01 **W = 1.229521123474350E+01** Tp=   260449.689197067026
 N = 4.092344516439835E+00 MA= 3.599930550170328E+02 **TA= 3.599892613233741E+02**
 A = 3.870983154191240E-01 AD= 4.661965567012716E-01 PR= 8.796913323250327E+01

 **w1= 12.295 deg**

 3182087.729166667 = **A.D. 4000-Feb-28 05:30:00.0000** TDB 
 EC= 2.060348142966012E-01 QR= 3.073425706719665E-01 IN= 6.886602138452942E+00
 OM= 4.578578064707545E+01 **W = 3.484244211886764E+01** Tp=  3182087.735363342799
 N = 4.092344820332250E+00 MA= 3.599746410639116E+02 **TA= 3.599606351364515E+02**
 A = 3.870982962554988E-01 AD= 4.668540218390312E-01 PR= 8.796912670002531E+01

 **w2= 34.842 deg**

 **w2-w1= 22.55 deg/8000.08 years = 1015 arcsec/100 years** (published value 574 arcsec/100 years)

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This result is essentially the same if a different period is chosen

Answer 1

You made three mistakes here, two major and one rather minor. These are

• Using $\arctan(y/x)$ in the Cartesian calculation.
• Using $\omega_2 - \omega_1$ in the orbital elements calculation.
• (Minor) Using 8000.08 years in both calculations as the span of time.

I'll address the last issue first. Astronomers use Julian years and Julian centuries. A Julian year is exactly 365.25 astronomical days long, where an astronomical day is exactly 86400 SI seconds long. There is no need to bother with leap years or leap seconds with this scheme. You should have used 7999.009012 years (or 79.99009012 centuries) instead of 8000.08 years.

The first two issues are related. Using $\arctan(y/x)$ as you did in the first calculation is not how the precession of Mercury is defined, nor is the change in the argument of perihelion, which is what you used in the second calculation.

Note that in the case of orbital elements that the right ascension of ascending node decreased from 55.58492230906509° to 45.78578064707545° over that 79.99 century span of time, or a nodal precession of -9.79914°. One needs to account for this nodal precession to obtain a full picture of Mercury's perihelion precession.

The standard approach is to use at the change in Mercury's longitude of perihelion, the sum of the right ascension of ascending node and the argument of perihelion. With this, $\Delta \bar\omega = (45.78578°+34.84244°)-(55.58492°+12.29521°)=12.74809°$. Dividing this by 79.99009012 centuries yields 0.159371°/century, or 573.735 arc seconds per century.