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