Post on 13-Dec-2015
Calculating MOIDs and using themfor estimating statistical impact
probabilities for terrestrial planets
Tomasz Wiśniowski, PhD studentSupervisor: Prof. Hans Rickman
Kraków 2013
Outline1.Developing our TOOLS
a) fast numerical computation of MOIDs• meridional plane method• last major improvements and new ideas
b) exploration of collisional zone
2. Achieving our GOALSa) estimating impact probability• MOID-chord method• MOID-track method• removing discrepancies, final comparisons and results
b) mapping of statistical impact probabilities • grid density vs. CPU time• results for Mars and Mercury
Quick reminder…
Quick reminder…
• not less accurate• faster • beter fitted for mapping purposes
The goal – to find competitive method
Quick reminder… TOOLS - Numerical method of MOID computation
Numerical method of MOID computationQuick reminder…
AL1 AM1 AR1STEP
BL1 BM1 BR1
BL2
BM2 BR2
AL2 AM2 AR2
BL2
DMIN2 DMIN1
TOOLS - Parallel tuning method
• high step of scanning & extreme speed (up to 12000 MOIDs per second )
• extreme accuracy (up to milimeters)
• interchange of accuracy and speed no more needed
• temporary results still possible
Results:
TOOLS - Parallel tuning method
• extreme accuracy uncovers some new missed MOIDs
FROM: Šegan, S., Milisavljevic, S., and Marceta, D. 2011, Acta Astron., 61, 275
Quick reminder… TOOLS – missed MOIDs problem
1. errors in scanning (~30 per 100 000)
Reasons of missing the MOIDs:TOOLS – missed MOIDs problem
2. „invisible” minima - intrinsic feature of meridional plane scanning (~1 per 100 000)
SOLUTION - looking for triplets instead of minima
SOLUTION – using „water” method
scanning with meridional plane
only one minimum?
initial tuning (all mimima)
final tuning (one mimimum)
„water”method
Y N
MOID
TOOLS – MOID computing graph
CHOISE
TOOLS – comparision of methods
TOOLS – website for computing the MOIDs
TOOLS – Exploring collisional zone
MOIDRcoll
Rcoll
MOID≤Rcoll
collisional zone
Vp
VT
Vp
VT VRRT
RG
VR
gravitational focusing
TOOLS – Calculating collisional radius
For any two orbits we can quickly:calculate the MOIDcalculate collisional radius Rcollanswer if collision is possible or notcalculate coordinates, velocities,
distances, movements, times etc. inside collisional zone
What is the probability of collision (impact probability) ?
TOOLS - SUMMARY
We know:• orbital and physical parameters of planetary target
GOALS – the problem approach
We generate for any point of the grid:• a huge number N of unperturbed projectiles with given ap,
qp, ip and random values of Ωp, ωp
We define:• the grid of orbital parameters ap, qp, ip of projectile
We calculate for any pair projectile-target:• probability of collision per orbit pcoll (=per one projectile’s
revolution)We calculate for any point of the grid:• mean statistical impact probability per orbit
We plot the map
pcoll probability of collision per orbit
= per one revolution of the projectile
GOALS - The probability of collision
Methods
• Wetherill’s (analytical)• Hill –sphere • MOID-chord • MOID-track
All methods give results in a good agreement
All methods were developed simultaneously
GOALS - The probability of collision
Vp
VT
Vp
VT VR
MOID
Rcoll
GOALS – MOID-chord method
We assume: The motion of projectile andtarget is uniform and rectilinear
Quick reminder…
MOID
α
VR
VRcosαRcoll
GOALS – MOID-chord methodQuick reminder…
Pt - period of the target
GOALS – MOID–track method
MOID RcollRcoll
t0t2 t1
time shift Δt= t2-t1
Pt - period of the target
We assume: No initial assumptions needed
meeting inside CZchasing inside CZ running towards each other to meet
(retrograde case)
How to calculate maximum target’s shift?
GOALS – MOID-track method
Δtmax=1321s
How to calculate maximum target’s shift?
1000s
collision!no collision!
100s 10s 1s1000s 100s 10s 1s
100s 10s100s
collision!MDAS(Δt):
MDAS(Δt) - Minimum Distance for Assumed ShiftMDAS(0)=MOIDMDAS(Δt) >Rcoll no collision !
MDAS(Δt) ≤Rcoll collision !
Δt
GOALS – MOID-track method
GOALS – impact probability – first results
November 2012October 2012
December 2012
ω
Ω
GOALS – mapping
GOALS – mapping
GOALS- Probability maps for Mars
GOALS- Probability maps for Mercury
ASSUMED ACCURACY:
• RMS error <1% (we need >100k collisions/point)
• linear approximation error <5%
• spline approximation error <???
GOALS- Grid density question
Thank you for your attention