Uncertainty parameter

The uncertainty parameter U is a parameter introduced by the Minor Planet Center (MPC) to quantify concisely the uncertainty of a perturbed orbital solution for a minor planet.[1][2] The parameter is a logarithmic scale from 0 to 9 that measures the anticipated longitudinal uncertainty[3] in the minor planet's mean anomaly after 10 years.[1][2][4] The uncertainty parameter is also known as condition code in JPL's Small-Body Database Browser.[2][4][5] The U value should not be used as a predictor for the uncertainty in the future motion of near-Earth objects.[1]

Orbital uncertainty

Classical Kuiper belt objects 40–50 AU from the Sun
Object JPL SBDB
Uncertainty
parameter
Horizons
January 2018
Uncertainty in
distance from the Sun
Refs
2013 BL76 1 ±40 thousand km JPL
20000 Varuna 2 ±140 thousand km JPL
19521 Chaos 3 ±840 thousand km JPL
(15807) 1994 GV9 4 ±1.4 million km JPL
(160256) 2002 PD149 5 ±8.2 million km JPL
1999 DH8 6 ±70 million km JPL
1999 CQ153 7 ±190 million km JPL
1995 KJ1 9 ±590 million km JPL
1995 GJ 9 ±160 trillion km JPL

Orbital uncertainty is related to several parameters used in the orbit determination process including the number of observations (measurements), the time spanned by those observations (observation arc), the quality of the observations (e.g. radar vs. optical), and the geometry of the observations. Of these parameters, the time spanned by the observations generally has the greatest effect on the orbital uncertainty.[6]

Objects such as 1995 SN55 with a condition code (Uncertainty Parameter U) of E where the eccentricity is assumed[7] are considered lost. 2010 GZ60 has an Uncertainty Parameter of 9, and could be an Earth threatening asteroid or may always remain beyond the asteroid belt.

Calculation

The U parameter is calculated in two steps.[1][8] First the in-orbit longitude runoff ${\displaystyle r}$  in seconds of arc per decade is calculated, (i.e. the discrepancy between the observed and calculated position extrapolated over ten years):

${\displaystyle r=\left(\Delta \tau \cdot e+10\cdot {\frac {\Delta P}{P}}\right)\cdot 3600\cdot 3\cdot {\frac {k_{o}}{P}}}$

with

 ${\displaystyle \Delta \tau }$ uncertainty in the perihelion time in days ${\displaystyle e}$ eccentricity of the determined orbit ${\displaystyle P}$ orbital period in years ${\displaystyle \Delta P}$ uncertainty in the orbital period in days ${\displaystyle k_{o}}$ ${\displaystyle 0.01720209895\cdot {\frac {180^{\circ }}{\pi }}}$ , Gaussian gravitational constant, converted to degrees

Then, the obtained in-orbit longitude runoff is converted to the "uncertainty parameter" U, which is an integer between 0 and 9. The calculated number can be less than 0 or more than 9, but in those cases either 0 or 9 is used instead. For instance: As of 10 September 2016, Ceres technically has an uncertainty of around −2.6, but is instead displayed as the minimal 0. The formula for cutting off the calculated value of U is

${\displaystyle U=\min \left(9,\max \left(0,\left\lfloor 9\cdot {\frac {\log r}{\log 648{,}000}}\right\rfloor +1\right)\right)}$

648 000 is the number of arc seconds in a half circle, so a value greater than 9 would mean that we would have basically no idea where the object will be in 10 years.

Function graph U(r)
U Runoff
0 < 1.0 arc second
1 1.0–4.4 arc seconds
2 4.4–19.6 arc seconds
3 19.6 arc seconds – 1.4 arc minutes
4 1.4–6.4 arc minutes
5 6.4–28 arc minutes
6 28 arc minutes – 2.1°
7 2.1°–9.2°
8 9.2°–41°
9 > 41°

References

1. ^ a b c d "Uncertainty Parameter U". Minor Planet Center. Retrieved 2011-11-15.
2. ^ a b c "Mission Design Center Trajectory Browser: Trajectory Browser User Guide". Ames Research Center. Retrieved 2016-03-03.
3. ^ "Editorial Notice" (PDF). The Minor Planet Circulars/minor Planets and Comets (MPC 24597–24780): 24597. 1995-02-15. Retrieved 3 March 2016.
4. ^ a b Drake, Bret G. (2011-01-01). "Strategic Implications of Human Exploration of Near-Earth Asteroids". NASA Technical Reports Server (NTRS). Retrieved 2016-03-03.
5. ^ "Definition/Description for SBDB Parameter/Field: condition code". JPL Solar System Dynamics. Retrieved 2011-11-15.
6. ^ "Near Earth Objects Close-Approach Uncertainties". NASA/JPL Near-Earth Object Program Office. 31 Aug 2005. Retrieved 2011-11-15.
7. ^ "Export Format for Minor-Planet Orbits". Minor Planet Center. Retrieved 2016-03-03.
8. ^ Desmars, Josselin; Bancelin, David; Hestroffer, Daniel; Thuillot, William (Jun 2011). Alecian, G.; Belkacem, K.; Samadi, R.; Valls-Gabaud, D. (eds.). "Statistical Analysis on The Uncertainty of Asteroid Ephemerides". SF2A 2011: Annual meeting of the French Society of Astronomy and Astrophysics. Paris, France: 639–642. Retrieved 3 March 2016.