Uncertainty parameter

{{Short description|Parameter introduced by the Minor Planet Center}}{{For|the quantification of uncertainty in general|Measurement uncertainty}}[[File:NEA-sizes-vs-uncertainties-compact.PNG|thumb|upright=1.25| The orbits of kilometre class NEAs are generally well known, though a few have been lost. However, large numbers of smaller NEAs have highly uncertain orbits.<ref name="MPC-NEA"/>]]

The '''uncertainty parameter''' ''U'' is introduced by the Minor Planet Center (MPC) to quantify the uncertainty of a perturbed orbital solution for a minor planet.<ref name="MPC-Uncertainty"/><ref name="User-Guide"/> The parameter is a logarithmic scale from 0 to 9 that measures the anticipated longitudinal uncertainty<ref name="MPC-24597"/> in the minor planet's mean anomaly after 10 years.<ref name="MPC-Uncertainty"/><ref name="User-Guide"/><ref name="Strategic-Implications"/> The larger the number, the larger the uncertainty. The uncertainty parameter is also known as '''condition code''' in JPL's Small-Body Database Browser.<ref name="User-Guide"/><ref name="Strategic-Implications"/><ref name="conditioncode"/> The ''U'' value should not be used as a predictor for the uncertainty in the future motion of near-Earth objects.<ref name="MPC-Uncertainty"/> == Orbital uncertainty == {| class="wikitable floatright" style="text-align:center; font-size:0.9em;" |+ Classical Kuiper belt objects 40–50 AU from the Sun |- style="align:center;vertical-align:bottom;" | JPL SBDB '''Uncertainty parameter'''   | Horizons January 2018 '''Uncertainty in distance from the Sun''' (millions of kilometers) | '''Object'''   | '''Reference Ephemeris''' Location: @sun Table setting: 39 |- ! 0 | 0.00 |align=left| (134340) Pluto | [https://www.minorplanetcenter.net/db_search/show_object?object_id=134340 E2022-J69] |- |- ! 1 | ±0.005 |align=left| 38628 Huya | {{JPL|38628}} |- ! 2 | ±0.03 |align=left| 19521 Chaos | {{JPL|19521}} |- ! 3 | ±0.10 |align=left| 15760 Albion | {{JPL|15760}} |- ! 4 | ±1.4 |align=left| {{mpl|(15807) Goibniu}} | {{JPL|15807}} |- ! 5 | ±8.2 |align=left| {{mp|(160256) 2002 PD|149}} | {{JPL|160256}} |- ! 6 | ±70 |align=left| {{mp|1999 DH|8}} | {{JPL|1999+DH8}} |- ! 7 | ±190 |align=left| {{mp|1999 CQ|153}} | {{JPL|1999+CQ153}} |- ! 8  | ±590 |align=left|{{mp|1995 KJ|1}} | {{JPL|1995+KJ1}} |- ! 9 | ±1,600 |align=left| 1995 GJ | {{JPL|1995+GJ}} |- ! ‘D’ |colspan="3;" align=left|   '''D'''ata insufficient for orbit determination. |- ! ‘E’ |colspan="3;" align=left|   '''E'''ccentricity was guessed instead of determined.<ref name="ExportFormat"/>

|- ! ‘F’ |colspan="3;" align=left|   Both ‘D’ and ‘E’ apply.<ref name="ExportFormat"/> |} 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.<ref name="NASA-uncertainty"/>

Occasionally, the Minor Planet Center substitutes a letter-code (‘D’, ‘E’, ‘F’) for the uncertainty parameter. :{| |- style="vertical-align: top;" ! D     | Objects with a ‘D’ have only been observed for a single opposition, and have been assigned two (or more) different designations ("double"). |- style="vertical-align: top;" ! E | Objects such as {{mp|2003 UU|291}} with a condition code ‘E’<ref name=IAU-MPC-2003-UU-291/> in the place of a numeric uncertainty parameter denotes orbits for which the listed '''e'''ccentricity was assumed, rather than determined.<ref name="ExportFormat"/> Objects with assumed eccentricities are generally considered lost if they have not recently been observed because their orbits are not well constrained.{{citation needed|date=April 2020|reason="Lost" not found in Export Format reference}} |- style="vertical-align: top;" ! F |Objects with an ‘F’ fall in both categories ‘D’ and ‘E’.<ref name="ExportFormat"/> |}

== Calculation == The ''U'' parameter is calculated in two steps.<ref name="MPC-Uncertainty"/><ref name="StatisticalAnalysis"/> First the in-orbit longitude runoff <math>r</math> in seconds of arc per decade is calculated, (i.e. the discrepancy between the observed and calculated position extrapolated over ten years):

:<math>r=\left(\Delta\tau\cdot e + 10\cdot\frac{\;\Delta P\;}{P}\right) \cdot 3600 \cdot 3 \cdot \frac{\;k_\text{o}\;}{P}</math>

with :{| |- | <math>\Delta\tau</math>|| uncertainty in the perihelion time in days |- | <math>e</math>|| eccentricity of the determined orbit |- | <math>P</math>|| orbital period in years |- | <math>\Delta P</math> || uncertainty in the orbital period in days |- | <math>k_\text{o}</math> || <math>0.01720209895\cdot\frac{180^\circ}{\pi}</math>, Gaussian gravitational constant, converted to degrees |}

Then, the obtained in-orbit longitude runoff is converted to the "uncertainty parameter" {{mvar|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. The formula for cutting off the calculated value of {{mvar|U}} is

:<math>U=\min \left\{ ~9, ~ \max \Bigl\{ \; 0, \; \left\lfloor 9\cdot\frac{\log r}{\;\log 648{,}000\;} \right\rfloor + 1 \; \Bigr\} ~ \right\} </math>

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 result is the same regardless of the choice of base for the logarithm, so long as the same logarithm is used throughout the formula; e.g. for "{{math|log}}" = {{math|log{{sub|10}}}}, {{math|log{{sub|''e''}}}}, {{math|ln}}, or {{math|log{{sub|2}}}} the calculated value of {{mvar|U}} remains the same if the logarithm is the same in both places in the formula.

frame|right|Function graph U(r) {| class="wikitable alternance centre" style="font-size:0.9em;" |+ |- ! scope="col" | {{mvar|U}} ! scope="col" | Runoff Longitude runoff per decade |- | 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° |}

648 000 is the number of arc seconds in a half circle, so a value greater than 9 would be meaningless as we would have no idea where the object will be in 10 years within the orbit. {{clear}}

== References == {{reflist|25em|refs=

<ref name="MPC-NEA"> {{cite web |title=Orbits for Near Earth Asteroids (NEAs) |department=Minor Planet Center |publisher=International Astronomical Union |url=https://www.minorplanetcenter.net/iau/MPCORB/NEA.txt |access-date=25 June 2020 |place= }} via {{cite web |title=M.P. Orbit Format |department=Minor Planet Center |publisher=International Astronomical Union |url=https://www.minorplanetcenter.net/iau/info/MPOrbitFormat.html }} </ref>

<ref name=IAU-MPC-2003-UU-291> {{cite web |title=2003 UU{{sub|291}} |department=Minor Planet Center |publisher=International Astronomical Union |url=http://www.minorplanetcenter.net/db_search/show_object?object_id=2003+UU291 }} </ref>

<ref name="MPC-Uncertainty"> {{cite web |title=Uncertainty parameter 'U' |department=Minor Planet Center |publisher=International Astronomical Union |url=http://www.minorplanetcenter.org/iau/info/UValue.html |access-date=2011-11-15 |df=dmy-all }} </ref>

<ref name="ExportFormat"> {{cite web |title=Export format for minor-planet orbits |department=Minor Planet Center |publisher=International Astronomical Union |url=http://www.minorplanetcenter.net/iau/info/MPOrbitFormat.html |access-date=2016-03-03 |df=dmy-all }} </ref>

<ref name="Strategic-Implications"> {{cite report |last=Drake |first=Bret G. |date=2011 |title=Strategic implications of human exploration of near-Earth asteroids |series=NASA Technical Reports |publisher=NASA |id=2011-0020788 |url=https://ntrs.nasa.gov/search.jsp?R=20110020788 |access-date=2016-03-03 |df=dmy-all }} </ref>

<ref name="User-Guide"> {{cite web |title=Trajectory Browser User Guide |series=Mission Design Center Trajectory Browser |publisher=NASA |department=Ames Research Center |url=http://trajbrowser.arc.nasa.gov/user_guide.php |access-date=2016-03-03 |df=dmy-all }} </ref>

<ref name="conditioncode"> {{cite web |title=Definition / description for SBDB parameter / field: condition code |department=JPL Solar System Dynamics |url=http://ssd.jpl.nasa.gov/sbdb_help.cgi?name=condition_code |access-date=2011-11-15 |df=dmy-all }} </ref>

<ref name="NASA-uncertainty">{{cite web |date=31 Aug 2005 |title=Near-Earth objects close-approach uncertainties |publisher=NASA / JPL |department=JPL Near-Earth Object Program Office |url=http://neo.jpl.nasa.gov/ca/neo_ca_info.html |url-status=dead |access-date=2011-11-15 |archive-url=https://web.archive.org/web/20040324140637/http://neo.jpl.nasa.gov/ca/neo_ca_info.html |archive-date=24 March 2004 |df=dmy-all }} </ref>

<ref name="MPC-24597"> {{cite report |title=Editorial Notice |series=The Minor Planet Circulars / Minor Planets and Comets |date=1995-02-15 |id=MPC 24597–24780 |pages=24597 |url=http://www.minorplanetcenter.net/iau/ECS/MPCArchive/1995/MPC_19950215.pdf |access-date=3 March 2016 }} </ref>

<ref name="StatisticalAnalysis"> {{cite journal |last1=Desmars |first1=Josselin |last2=Bancelin |first2=David |last3=Hestroffer |first3=Daniel |last4=Thuillot |first4=William |date=Jun 2011 |title=Statistical analysis on the uncertainty of asteroid ephemerides |url=http://hal.upmc.fr/hal-00647644 |editor1-last=Alecian |editor1-first=G. |editor2-last=Belkacem |editor2-first=K. |editor3-last=Samadi |editor3-first=R. |editor4-last=Valls-Gabaud |editor4-first=D. |journal=SF2A 2011: Annual Meeting of the French Society of Astronomy and Astrophysics |pages=639–642 |location=Paris, France |bibcode=2011sf2a.conf..639D |access-date=3 March 2016 }} </ref>

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Category:Orbits Category:Measurement