<p>To compute asteroid's position you can use NASA  <a href="https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/" rel="nofollow noreferrer">Small-Body Database Lookup</a>  web page for orbital data, ephemeris and uncertainties.
To check the calculated positions you can use  <a href="https://ssd.jpl.nasa.gov/horizons/" rel="nofollow noreferrer">JPL's Horizon system</a>  for more options.</p>
<p>example: <a href="https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2000%20FX40" rel="nofollow noreferrer">2000 FX40(id 20050835)</a> or <a href="https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2014%20HO64" rel="nofollow noreferrer">2014 HO64 (id 54099790)</a></p>
<p>Note that the first is a numbered asteroid with more precise orbital data (12 km) than the second (2040 km)</p>
<p>Image of their orbits and point clouds looking from
Z axis</p>
<p><a href="https://i.sstatic.net/TppDl1iJ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/TppDl1iJ.png" alt="2000 FX40 &amp; 2014 HO64" /></a></p>
<p>This approach at 300 km will happen 2024-09-11 04:55 UTC.
Check <a href="https://ssd.jpl.nasa.gov/api/horizons.api?format=text&amp;COMMAND=%27DES=2014%20HO64%27&amp;START_TIME=%272024-09-10%2005:02%27&amp;STOP_TIME=%272024-09-12%2005:02%27&amp;STEP_SIZE=%2748%27&amp;CENTER=%27301@399%27&amp;MAKE_EPHEM=%27YES%27&amp;EPHEM_TYPE=%27OBSERVER%27&amp;QUANTITIES=%271,2,9,20,23,24,25%27&amp;CSV_FORMAT=%27YES%27&amp;ANG_FORMAT=%27DEG%27%20" rel="nofollow noreferrer">viewing conditions at specific sites and time</a> with Horizon's system.
It would be interesting to have a more precise orbit for (2014 HO64). Many more observations are required.</p>
<p>PS: link for photography of 2000 FX40 &amp; 2014 HO64 approach by <a href="https://groups.io/g/mpml/message/39872" rel="nofollow noreferrer">Filipp Romanof posting on mpml@groups.io</a></p>
<p><a href="https://i.sstatic.net/V08XInyt.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/V08XInyt.jpg" alt="enter image description here" /></a></p>
<p>or this link for <a href="https://groups.io/g/mpml/topic/2000_fx40_and_2014_ho64_close/108314282" rel="nofollow noreferrer">further info on the whole thread</a>.</p>

CC BY-SA 4.0
2024-09-06T14:31:21.880Z
2024-09-11T17:57:43.677Z
2024-09-11T17:57:43.677Z
nadowe
<p>A few years ago I made a program to <a href="https://astronomy.stackexchange.com/questions/43293/asteroids-conjunction/45468#45468">compute asteroids positions and focus on close approaches</a> (under 10 000 km). On average there is one every day. Leftmost numbers represent <a href="https://astronomy.stackexchange.com/questions/51090/how-does-one-convert-between-modified-julian-date-mjd-and-a-standard-mm-dd-yr">MJD date</a> and 'ds' is distance in 1000 km between asteroids.</p>
<p><a href="https://i.sstatic.net/4hMUEdXL.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4hMUEdXL.jpg" alt="enter image description here" /></a></p>
<p>Now it would be interesting to know the probability that the asteroids are actually at the predicted positions.
Thanks to <a href="https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/" rel="nofollow noreferrer">JPL Small-Bodies Database</a> we know the uncertainties of all osculating orbital elements.</p>
<p>Given those uncertainties, how can we calculate the radius of a blurred sphere representing an asteroid's position with a given precision, let say 78% which correspond to JPL's published uncertainties of one-Sigma?</p>
<p>Following Greg Miller's suggestion I calculated position varying orbital data within 6 embedded loop with zero, plus and minus uncertainty for each of the 6 osculating elements.</p>
<p>This give 3^6 or 729 positions. But the resulting points cloud seems quite unrealistic, not close to a sphere but an elongate cigar! Like this picture of 2 asteroid in close approach.</p>
<p><a href="https://i.sstatic.net/OwHiJr18.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OwHiJr18.png" alt="enter image description here" /></a></p>
<p>What am I doing wrong?</p>

CC BY-SA 4.0
2024-08-24T14:28:23.930Z
Choose random numbers +/- the uncertainty and use it to compute the position.  Repeat a bunch of times to create a point cloud.
Greg Miller
CC BY-SA 4.0
2024-08-24T15:26:39.440Z
Great solution! Sure must work. But does an analytic solution exist?
nadowe
CC BY-SA 4.0
2024-08-25T02:17:09.047Z
This is an interesting question, but I think there is a problem here, having to do with [correlations between parameters](https://stats.stackexchange.com/questions/tagged/correlation) and how those uncertainties are *defined* and calculated.
uhoh
CC BY-SA 4.0
2024-08-25T02:19:54.720Z
But I have a question - what does your program actually DO? For example, do you obtain the osculating parameters for the specific time that you calculate the position, or do you use osculating parameters from some other time, and then propagate a Keplerian orbit based on those parameters forwards or backwards in time to calculate closest approach. I think tyou need to make that clear before a correct, useful answer providing an "analytical solution" can be provided. *Thanks!*
uhoh
CC BY-SA 4.0
2024-08-25T11:57:51.820Z
Actually, I do not propagate Keplerian orbit. I used the periodically published elements (last epoch is MJD 60600) calculate position for main belt asteroid keeping track of candidates within 10 000 km then check against JPL Horizon system for more accuracy.
nadowe
CC BY-SA 4.0
2024-09-05T16:46:07.820Z
I would not expect the point cloud to resemble a circle, the ellipses you got seem plausible.  The errors will generally not be the same in all dimensions, which gives the elliptical shape.
Greg Miller
CC BY-SA 4.0
2024-09-06T13:09:16.620Z
Ok, Greg, I realize that's correct.
Now there is the correlations between parameters like uhoh pointed out that might distort the cloud shape. I am using the semi-major axis to compute the period of rotation and determine mean anomaly. So using uncertainties of axis AND anomaly might be overdoing thing.
nadowe
<asteroids><orbital-elements>
How to plot asteroid position according to uncertainties on osculating orbital elements?
nadowe
# How to plot asteroid position according to uncertainties on osculating orbital elements?

(Asked by: username_0 on 2024-08-24)

A few years ago I made a program to [compute asteroids positions and focus on close approaches](https://astronomy.stackexchange.com/questions/43293/asteroids-conjunction/45468#45468) (under 10 000 km). On average there is one every day. Leftmost numbers represent [MJD date](https://astronomy.stackexchange.com/questions/51090/how-does-one-convert-between-modified-julian-date-mjd-and-a-standard-mm-dd-yr) and 'ds' is distance in 1000 km between asteroids.

[![enter image description here](https://i.sstatic.net/4hMUEdXL.jpg)](https://i.sstatic.net/4hMUEdXL.jpg)

Now it would be interesting to know the probability that the asteroids are actually at the predicted positions.
Thanks to [JPL Small-Bodies Database](https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/) we know the uncertainties of all osculating orbital elements.

Given those uncertainties, how can we calculate the radius of a blurred sphere representing an asteroid's position with a given precision, let say 78% which correspond to JPL's published uncertainties of one-Sigma?

Following username_1 username_1's suggestion I calculated position varying orbital data within 6 embedded loop with zero, plus and minus uncertainty for each of the 6 osculating elements.

This give 3^6 or 729 positions. But the resulting points cloud seems quite unrealistic, not close to a sphere but an elongate cigar! Like this picture of 2 asteroid in close approach.

[![enter image description here](https://i.sstatic.net/OwHiJr18.png)](https://i.sstatic.net/OwHiJr18.png)

What am I doing wrong?

**Tags:** asteroids, orbital-elements

> **Comments:**
> **username_1**: Choose random numbers +/- the uncertainty and use it to compute the position.  Repeat a bunch of times to create a point cloud.  
> **username_0**: Great solution! Sure must work. But does an analytic solution exist?  
> **username_2**: This is an interesting question, but I think there is a problem here, having to do with [correlations between parameters](https://stats.stackexchange.com/questions/tagged/correlation) and how those uncertainties are *defined* and calculated.  
> **username_2**: But I have a question - what does your program actually DO? For example, do you obtain the osculating parameters for the specific time that you calculate the position, or do you use osculating parameters from some other time, and then propagate a Keplerian orbit based on those parameters forwards or backwards in time to calculate closest approach. I think tyou need to make that clear before a correct, useful answer providing an "analytical solution" can be provided. *Thanks!*  
> **username_0**: Actually, I do not propagate Keplerian orbit. I used the periodically published elements (last epoch is MJD 60600) calculate position for main belt asteroid keeping track of candidates within 10 000 km then check against JPL Horizon system for more accuracy.  
> **username_1**: I would not expect the point cloud to resemble a circle, the ellipses you got seem plausible.  The errors will generally not be the same in all dimensions, which gives the elliptical shape.  
> **username_0**: Ok, username_1, I realize that's correct. Now there is the correlations between parameters like username_2 pointed out that might distort the cloud shape. I am using the semi-major axis to compute the period of rotation and determine mean anomaly. So using uncertainties of axis AND anomaly might be overdoing thing.  

---

## 1 answers

### Answer by username_0 on 2024-09-06. Score: 2

To compute asteroid's position you can use NASA [Small-Body Database Lookup](https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/) web page for orbital data, ephemeris and uncertainties.
To check the calculated positions you can use [JPL's Horizon system](https://ssd.jpl.nasa.gov/horizons/) for more options.

example: [2000 FX40(id 20050835)](https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2000%20FX40) or [2014 HO64 (id 54099790)](https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2014%20HO64)

Note that the first is a numbered asteroid with more precise orbital data (12 km) than the second (2040 km)

Image of their orbits and point clouds looking from
Z axis

[![2000 FX40 & 2014 HO64](https://i.sstatic.net/TppDl1iJ.png)](https://i.sstatic.net/TppDl1iJ.png)

This approach at 300 km will happen 2024-09-11 04:55 UTC.
Check [viewing conditions at specific sites and time](https://ssd.jpl.nasa.gov/api/horizons.api?format=text&COMMAND=%27DES=2014%20HO64%27&START_TIME=%272024-09-10%2005:02%27&STOP_TIME=%272024-09-12%2005:02%27&STEP_SIZE=%2748%27&CENTER=%27301@399%27&MAKE_EPHEM=%27YES%27&EPHEM_TYPE=%27OBSERVER%27&QUANTITIES=%271,2,9,20,23,24,25%27&CSV_FORMAT=%27YES%27&ANG_FORMAT=%27DEG%27%20) with Horizon's system.
It would be interesting to have a more precise orbit for (2014 HO64). Many more observations are required.

PS: link for photography of 2000 FX40 & 2014 HO64 approach by [Filipp Romanof posting on mpml@groups.io](https://groups.io/g/mpml/message/39872)

[![enter image description here](https://i.sstatic.net/V08XInyt.jpg)](https://i.sstatic.net/V08XInyt.jpg)

or this link for [further info on the whole thread](https://groups.io/g/mpml/topic/2000_fx40_and_2014_ho64_close/108314282).


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