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Modelling circumplanetary ejecta clouds at low altitudes: a probabilistic approach
Apostolos A. Christoua,
a



Armagh Observatory, Col lege Hil l, Armagh BT61 9DG, Northern Ireland, UK

Abstract A mo del is presented of a ballistic, collisionless, steady state population of ejecta launched at randomly distributed times and velo cities and moving under constant gravity above the surface of an airless planetary bo dy. Within a probabilistic framework, closed form solutions are derived for the probability density functions of the altitude distribution of particles, the distribution of their speeds in a rest frame both at the surface and at altitude and with respect to a moving platform such as an orbiting spacecraft. These expressions are validated against numerically-generated synthetic populations of ejecta under lunar surface gravity. The mo del is applied to the cases where the ejection speed distribution is (a) uniform (b) a power law. For the latter law, it is found that the effective scale height of the ejecta envelope directly depends on the exponent of the power law and increases with altitude. The same holds for the speed distribution of particles near the surface. Ejection mo del parameters can, therefore, be constrained through orbital and surface measurements. The scope of the mo del is then extended to include
Corresponding author Email addresses: aac@arm.ac.uk (Apostolos A. Christou), Fax: (Apostolos A. Christou)


+44 2837 527174

Preprint submitted to Icarus

December 15, 2014


size-dependency of the ejection speed and an example worked through for a deterministic power law relation. The result suggests that the height distribution of ejecta is a sensitive proxy for this dependency. Keywords: Dust, Dynamics, Impacts, Mo on 1. Introduction The surfaces of the Mo on and other airless bo dies in the solar system are sub ject to continuous bombardment by interplanetary meteoroids (Grun ¨ et al., 1985) resulting in their being immersed in continuously-replenished clouds of impact ejecta (Gault et al., 1963). Such clouds have been detected in situ around the Galilean satellites and at distances of a few tenths of satellite radii or further away (Kruger et al., 2000, 2003). For the case of ¨ the Mo on there has been indirect evidence for two distinct dust populations, one at altitudes 100km (McCoy and Criswell, 1974; McCoy, 1976) and the

other at some centimetres to hundreds of m above the surface (Rennilson and Criswell, 1974; Severny et al., 1975; Berg et al., 1976). The mobilising action of electrostatic forces on surface dust grains has been involved in both cases (Criswell, 1973; Rhee et al., 1977; Zo ok and McCoy, 1991; Stubbs et al., 2006). Mo delling of the ensemble properties of meteoroid impact ejecta specifically at these low altitudes 0.1 radii above the surface - and their

contribution to the flux of near-surface particles has received comparably little attention. Although mo dels that are formally valid for any distance do exist (Krivov et al., 2003; Sremevic et al., 2003), the fo cus has been on c repro ducing measurements at higher altitudes. In particular, Krivov et al. (2003) found that the radial dependence of dust density may be approximated 2


by a power law with an exponent of -5/2 at distances of several satellite radii or larger. Data from the recent LADEE mission to the Mo on are expected to shed new light on the dust environment around airless bo dies (Elphic et al., 2011) and justifies detailed mo delling of this low-altitude component of the ejecta cloud. In this paper, we mo del this cloud as a steady state pro cess and show how, under certain assumptions valid at low altitude, it is possible to derive explicitly the probability distributions of many of its properties for arbitrary ejection speed laws. Our aim is three-fold: (i) to fill a notable gap in the literature on analytical mo delling of circumplanetary ejecta clouds in the low-altitude regime, (ii) to prepare for the interpretation of new datasets from current and future missions as well as to revisit existing datasets with new mo delling to ols, and (iii) to describe and demonstrate a general framework within which more sophisticated mo dels can be constructed. The paper is organised as follows: In the next Section we describe the kinematics of one-dimensional ballistic particle motion following vertical ejection from the surface. The regime where a constant gravitational acceleration can be considered a realistic approximation is also quantified here. Section 3 describes the steady state cumulative distribution and probability density functions (pdfs) of particle altitude for a given value of the (vertical) ejection speed. A metho dology is then presented which uses these expressions in deriving the altitude pdfs for arbitrary - but size-independent - distributions of ejection speed. Explicit formulae for the altitude pdfs both for a uniform and a power-law probability distribution of ejection velo cities are derived. In Section 4 we derive pdfs for the speed at a given altitude both in a rest frame and with respect to a moving platform. In Section 5 we lift the assumption

3


of vertical ejection and consider the motion of ejecta in three dimensions. Section 6 intro duces size-dependency to the physical pro cess of ejection. An expanded formalism is intro duced to treat this class of problems and its use is demonstrated by way of an example. The last Section summarises our findings, discusses mo del limitations and the scope of this approach for the interpretation of in situ dust measurements and suggests several avenues for future work. 2. Theoretical framework and relation to previous works Previous published mo dels of dust clouds are expressed in terms of physical quantities (eg. the grain number density as a function of with altitude) and utilise the full equations of motion for Newtonian gravity. In their mo del, Kruger et al. (2000) assumed vertical ejection of grains; Krivov et al. ¨ (2003) lifted that assumption by assuming a distribution of ejection directions. Sremevic et al. (2003) extended the treatment to the asymmetric c distribution of ejecta surrounding a satellite in orbit around a planet under an isotropic flux of impactors. Here, we cho ose to investigate the properties of the cloud within a probabilistic framework in the sense that the derived probability density functions integrate to unity. Since our fo cus is on the near-surface regime, we assume constant gravitational acceleration g = GM /R2 where G is the Newtonian gravitational constant and M , R are the mass and radius of the bo dy respectively. A consequence of this choice is that the position variable in our mo del is the altitude h above the surface rather than the distance r from the bo dy's centre of mass. For vertical

4


ejection velo city vL at ejection time tL the kinematic equations are 1 v = vL - g (t - tL ), h = vL (t - tL ) - g (t - tL ) 2
2

(1 )

where v and t are, respectively, the in-flight velo city and time. From these relations one finds that the maximum altitude hm tm
ax ax

and corresponding time

a re hm
ax 2 = vL /2g , tm ax

= vL /g

(2 )

with respect to the moment of ejection. The kinematic equation for the altitude h can then be written 1 h = hmax - g (t - tmax )2 . 2 (3 )

It is useful here to know the regime where the above expressions are a go o d approximation to the full Kepler problem i.e. where the gravitational acceleration varies as a function of r . We do this below by comparing the value of the maximum altitude achieved by a particle in the two cases. From conservation of energy, we have v2 = µ (2/r - 1/a) (4 )

where a the semima jor axis of the keplerian ellipse and µ = GM . The semima jor axis of this ellipse may be calculated by setting v = vL and r = R. By setting v = 0, we can then solve for r the bo dy r
max 2 = R/ 1 - vL R/2µ . max max

, the maximum distance from

(5 )

2 If = vL R/2µ 1 we can write r

pression for hm

ax

(Eq. 2 above) for g = µ/R2 . The relative error hmax /hm 5

R(1 + ), recovering the exax


incurred by omitting the additional terms in the series is O (). For the case of the Mo on that is relevant to the following Sections, we can assume R = 1738 km and µ = 4908 km3 sec-2 . Requiring that < 0.1 then imposes the limit vL < 0.75 km sec-1 . This corresponds to a maximum altitude of 190 km above the lunar surface, implying that for h 200 km the approx-

imate formulas (1)-(3) may be considered as a go o d approximation for the lunar case. Note that the stated limiting ejection velo city is lower than the escape velo city from the lunar surface, 2.4 km sec-1 , implying that our mo del ignores grains not gravitationally bound to the Mo on. In the works cited at the beginning of this Section it was shown (cf. Panel a, Figure 9 of Kruger et al., 2000) that the contribution of these escaping grains is minimal ¨ near the surface. Therefore, they can be safely neglected here. A further working assumption in our mo del is a "flat surface" approximation where the volume of space above the surface increases as a linear, rather than a quadratic, function of altitude. This incurs an error of order (h/R)2 10
-2

for the limiting value of h stated above1 . Finally, we assume that all grains are ejected at the same angle from the surface normal. In this sense, the scope of our mo del is intermediate between those of Kruger et al. (2000) ¨ and Krivov et al. (2003) in that we consider non-vertical, as well as vertical, ejection but not distributions of ejection angles.
1

An alternative formulation of this assumption is that two slabs of equal thickness h

have equal volume. In the full problem, these slabs map into spherical shells with a volume proportional to r = R + h

6


3. Distributions of particle altitude In notation used here and throughout the paper, the capital symbol (e.g. H ) denotes a random variable, the probability distribution of which is sought, while the corresponding small case symbol (h) refers to the particular value adopted for that variable. The cumulative distribution function (cdf ) of the altitude h can be arrived at with the help of Eq. 3 by considering the fraction of the total time-of-flight (2tmax ) spent above h: F (h) = P (H < h) = 1 - 1 - h/hmax , 0 h < hm
ax

(6 )

with the corresponding probability density function (pdf ) being obtained by differentiation: f (h) = P (h < H < h + dh) = 1/ 2 hmax (hmax - h) , 0 < h < hmax . (7)

We note that the expectation of this probability density is E [h] = (2/3) hmax . Formally, to derive the probability P in Eq. 7 one would still need to multiply the function f (h) with the appropriate differential (dh in this case) but we opted to omit this throughout the paper to keep the expressions as simple as possible (but hopefully not simpler!). We now assume a Poissonian pro cess of particles being ejected from the surface with a rate parameter . It can be shown that, if 1/ tm particular value of vL . As an independent verification of the above expressions, we have generated N = 104 uniformly distributed random variates of launch time within the interval [0, 2tmax ], calculated the respective altitudes using Eq. 3 at some 7
ax

then

Eq. 7 describes the steady-state altitude distribution of particles for that


random time with respect to the start of the simulation, plotted the resulting probability density and cumulative distributions and superposed the curves described by Eq. 7 and Eq. 6 respectively. The result is shown in Fig. 1 where we have set g = 1.6 m sec parameter is (1/2) 104 /tm
ax -2

and vL = 180 m sec-1 . The effective rate
-1

45 sec

and tm

ax

= 112 sec so the requireax

ment for a steady state is satisfied. Note the cusp at h = hm

due to the

denominator of the pdf in Eq. 7 and its first derivative vanishing for that value of the altitude. Next, we allow the ejection speed to vary according to a probability density function g (vL ). The resulting steady-state pdf for the altitude at a given time t may be evaluated as: fV (h) = N (h < H < h + dh)/N (H > 0) where N (H > 0 ) = and N (h < H < h + dh) = g (vL )f (h)dtL dvL . (1 0 ) g (vL )dtL dvL (9 ) (8 )

The integration over tL = t - tL is due to the fact that only particles that are in flight (ie tL < 2tmax ) can contribute to the pdf. For example, let us assume a uniform ejection speed distribution
- + + - g (vL ) = P (vL < VL < vL + dvL ) = 1/ vL - vL , vL vL vL .

(1 1 )

Eq. 11 is not, strictly speaking, realistic in the sense that it is not discussed or considered in the literature in relation to mo delling ejecta populations. It is inserted here mainly as a paedagogical to ol to demonstrate the 8


metho dology advo cated in this work and to highlight the emergence of certain features as one transitions from the monotachic2 (single value of ejection speed) regime to a distribution of ejection speeds for the particles. As shown later in the paper, such features are also observed when more physically relevant ejection speed mo dels are adopted. Upon double integration, Eq. 9 yields
+ - N (H > 0) = (/g ) vL + vL

(1 2 )

2 2g h/vL. It becomes

Expression 10 can be evaluated using the transformation cos2 S = 1 -

+ - N (h < H < h + dh) = 2/ vL - vL

sin-1 S dS

(1 3 )

Substituting these into Eq. 8 then yields the altitude pdf
+ - fU (h) = -2g Bu vL , h - Bu max{vL ,

2g h}, h

+ - / vL 2 - vL

2

(1 4 )

where Bu (vL , h) = log tan arcsin
2 2g h/vL /2

(1 5 )

and the subscript "U" refers to the uniform speed distribution assumed. To verify this result, we utilise uniform random variates of launching time as before. N = 105 variates are generated within the fixed time interval [0, 1000] sec for a rate parameter of 100 sec-1 . We assign to each particle a launch speed value randomly drawn from the distribution given by Eq. 11
- and for three different choices of the defining parameters: (a) vL = 490 m

2

From the greek = velocity ´

9


+ sec-1 , vL = 500 m sec

-1

- + (b) vL = 300 m sec-1 , vL = 500 m sec-1 , and

- + (c) vL = 10 m sec-1 , vL = 500 m sec-1 . We note for future reference + + that tmax (vL ) = 312.5 sec while hmax (vL ) = 78 km. The results of the

simulation and at t = 700 sec are shown in the left panels of Fig. 2 where we have overplotted the analytical solution (black curve). For case (a), the resulting distribution is similar to that for a constant ejection speed, as can
- be intuitively expected. For case (b), a maximum o ccurs near h = hmax (vL )

(indicated by the white arrow). That, and the cusp seen in the analytical - laws, are due to the fact that particles launched with vL < vL < 2g h (Eq. 2)
- no longer contribute in Eq. 14 as h increases beyond the value hmax (vL ).

An ejection speed law relevant to mo delling ejecta pro duction from satellite surfaces (Krivov, 1994; Kruger et al., 2000; Krivov et al., 2003; Kruger ¨ ¨ et al., 2003) has the form p(vL ) = ( /v0 ) (vL /v0 ) where v0 and are positive constants. Following the same pro cedure as above we obtain N (H > 0) = 2 (v0 /g ) / ( - 1) and
N (h < H < h + dh) = -2 v0 (2g h)- 2 where cos2 S = 1 - 2g h/ (u2 v0 ), u = vL /v0 . ( +1)/2 0 - -1

, vL > v0

(1 6 )

(1 7 )

sin S dS
S|
u= 1

(1 8 )

Substituting in Eq. 8 and evaluating the integral yields fP (h) = g ( -1)(2g h)-
( +1)/2 -1 v0

Bp (1) -Bp max{ 10

2 1-2g h/v0 , 0}

(1 9 )


where Bp (x) = x 2 F1 1/2, (1 - )/2, 3/2, x2 ,
2

(2 0 )

F1 (.) denoting the Gauss hypergeometric function and the subscript "P"

referring to the power law assumed for the ejection speed distribution. The right panels of Fig. 2 show these analytical expressions overplotted on the result of a numerical simulation where N = 2 â 106 variates are generated

within the fixed time interval [0, 4000] sec for a rate parameter of 500 sec-1 . The launch speed value was randomly drawn from the distribution given by Eq. 16 with = 1.2 and for three different choices of v0 : (a) v0 = 1 m sec-1 , (b) v0 = 10 m sec
-1

and (c) v0 = 50 m sec-1 . The altitude statistics shown

were collected at t = 2000 sec. To speed up the Monte Carlo simulations we imposed a cutoff value for the ejection speed of vcutoff = 800 m sec-1 . This changes slightly the mo del to the effect that the expression for N (H > 0) needs to be multiplied by a factor of 1 - (vcutoff /v0 )
1-

. Formally, one also

needs to change the upper limit of the integral in Eq. 18; however, since most of the power is in the small end of the speed spectrum, the result remains essentially unchanged. Certain properties of the altitude distribution are worthy of further comment. A cusp, seen for the case of the uniform speed distribution (left panels), is also present here and corresponds to h = hmax (v0 ) (indicated by the arrow). In the top panel of Fig. 3, Eq. 19 is plotted for different values of v0 : 1, 2, 5 and 10 m sec-1 . The scale in both axes is logarithmic. There we see that P
max

= P (hmax ) is consistently higher than the value at the surface

P0 (= P (h = 0)). As v0 increases, these both decrease and the power under P (h) for h < hmax gradually spreads to higher altitudes. Formally, one can 11


trace these features to the expressions for P0 and P parameters g , v0 and : P0 = g ( - 1) ,P 2 v0 ( + 1)
max

max

in terms of the mo del

=



P

0

( + 1 )

+1 2 2

(2 1 )

where denotes the Gamma function. Observe that both probabilities are
2 inversely proportional to v0 and that the ratio of P max

/P0 depends only on the

speed exponent and is greater than unity 3 . On the far side of the cusp, the 2 same value of P as P0 is attained at h = (v0 /2g ) [( /2) ( + 1) ( 1+ )/(1+ 2
2

)]2/(

+1)

, bounding the near-surface region with the highest number density.

For h > hmax the profiles are linear in log-log space ie log P (h)/ log h is constant. This follows from Eq. 19 where the quantity in square brackets becomes independent of h so taking the log of both sides results in a linear law with slope -( + 1)/2. A consequence of this property is that
dP dh

/P

decreases with increasing altitude (bottom panel of Fig. 3). In other words, the "effective scale height" i.e. the inverse of the plotted quantity, is altitudedependent and increases as h increases. As a final note in this Section, we show how the functional form of Eq. 19 can be retrieved by imposing the assumptions of Section 2 on the mo del by Krivov et al. (2003). In that work, particles are ejected within a cone of opening half-angle 0 (their Fig. 3) and with a distribution of ejection angles given by their Eq. 21. Their results and those in this Section should agree for the case 0 = 0 (vertical ejection of grains). Using their notation we find u2 v 2 + h/rM to first order in h/rM , then set 1 - (h/rm ) = cos2 S and ~ ~ change the dummy variable in their Eq. 44 from u to S . The - ( + 1) /2 ~
3

For 0 < < 3, Pm

ax

> 0.9 P0

12


power-law dependence on h is obtained from their Eq. 42 as the zero-order term in the power-series expansion of r -2 in terms of h/rM . ~ 4. Distributions of particle speed at a given altitude Another important descriptor of the ejecta population is the distribution of speed v at a given altitude h. The pdf of this distribution may be expressed as P (V = v|H = h) = P (V = v, H = h)/P (H = h) (2 2 )

where P (V = v, H = h) is the joint probability density of particles with speeds between v and v + dv and altitudes between h and h + dh. The reader is reminded that, in the notation of conditional probability, the quantities on the right-hand side of the separator " | " are treated as constant parameters while those on the left-hand side denote random variables. Considering the transformation (vL , tL = t - tL ) (h, v) we can write P (VL = vL (h, v), TL = tL (h, v))||J ||dvdh = P (VL = vL , TL = tL )dtL dvL (2 3 )

where the right-hand side represents the number of particles launched between tL and tL + dtL ago and with launch speeds between vL and
- value of its determinant. From Eq. 1 we find ||J || = vL 1 (v, h) while the joint

vL + dvL , J is the Jacobian of the transformation and ||.|| denotes the absolute

density distribution for tL and vL can be expressed as P (VL = vL , TL = tL ) = P (TL = tL |VL = vL )Pt (VL = vL ) (2 4 )

13


For a steady state pro cess, the first term on the right hand side of Eq. 24 is the recipro cal of the time-of-flight, 2vL /g . The second term is the probability that a particle in flight at time t was launched with speed vL . For a uniform ejection speed distribution, the latter probability is proportional to the time-of-flight. The constant of proportionality may be found by integrating the probability over all possible values of vL . The joint probability density of vL and tL is then P (VL = vL , TL = tL ) = g / v and from Eq. 23 we find P (V = v, H = h) = g / v
+2 +2

-v

-2

(2 5 )

-v

-2

v 2 + 2g h

(2 6 )

Making use of Eq. 14 in Section 3, Eq. 22 then becomes P (V = v|H = h) = -(1/ v
-2

v2 + 2g h)/ 2g h}, h , (2 7 )

Bu v+ , h - Bu max{v- , - 2g h < v < v
+2

- 2g h

where the quantity Bu is from Eq. 15. For the power law distribution given by Eq. 16, pro ceeding along similar lines yields P (V = v|H = h) = (2g h)
( +1)/2 +2)/2

(v2 + 2g h)(

/
2 max{1 - 2g h/v0 , 0}

Bp (1) - Bp v>
2 v0 - 2g h

, (2 8 )

14


where the quantity Bp is now that of Eq. 20. For this case, the second
- term in Eq. 24 is of the form Pt vL â vL -1

.

In Fig. 4 we show the analytical expressions for the uniform (left panels) and the power law (right panels) distributions compared with speed statistics of N = 107 particles at altitudes - from top to bottom - of 10 m, 100 m, 1000 m and 10 km. The parameters of the uniform distribution were set at v- = 30 m sec
-1

and v+ = 200 m sec-1 ; for the power-law distribution,

was set to 1.2 and v0 to 10 m sec-1 . Two features are worthy of note: (i) the distributions at lower altitudes are truncated at some non-zero value of v due to the existence of a minimum value for vL , and (ii) the power under the distributions for the power-law ejection speed gradually spreads to a wider range of v values, with the result that particles with speeds in excess of 100 m sec
-1

are unlikely for h 1000m but as likely as those with

speeds below that value for h = 10km. In any case, most of the particles would be concentrated at the lower altitudes, as the middle right panel of Fig. 2 indicates. The spreading of the power towards higher speeds with increasing altitude may be seen more clearly in the top panel of Fig. 5 where we have plotted Eq. 28 for altitudes starting from 10 m above the surface (black curve) and increasing, by factor-of-10 increments, to 105 m (100 km; light grey curve). In particular, particle speeds 100 m sec km the existence of particles with any speed up to 500 m sec probable. In the bottom panel of the same Figure one sees the same expression plotted as above for h =1 m up to h =1000 m but on a logarithmic speed
-1

are an order

of magnitude more probable at 1000 m than at 10 and 100 m while at 100
-1

is comparably

15


scale. This shows that, at the lowest altitudes (e.g. h = 1m), the conditional probability density of speed given the altitude is approximately a powerlaw function of speed. At higher altitudes, it is insensitive to speed up to v 2g h, becoming a power law at higher speeds. This follows from Eq. 28 where, if v2 2g h then P (V = v|H = h) is approximately equal to (2g h)-
-( +2) 1/2

while, when the situation is reversed, P (V = v|H = h)

v

. Therefore, as found for the altitude pdf examined in Section 3,

the distribution of particle speeds depends on the ejection physics with the dependence being more apparent near the surface. In relation to the in situ characterisation of ejecta clouds, the frame in which measurements are collected is not always inertial. It is instructive, therefore, to consider the pdf of the speed w of grains relative to a platform moving horizontally with speed u (Fig. 7). In this case, we have P (W = w|H = h) = P (V = v (u, w) |H = h) w/v (u, w) where v (u, w) = w2 - u2 . (2 9 )

In Fig. 6 we compare the analytical expression (29) with the results of numerical simulations and the following choices of parameter values: uniformlydistributed ejection velo cities at h = 1000 m and a platform moving with u = 1650 m sec
-1

(top left); at h = 100 m and u = 100 m sec

-1

(b o t-

tom left); power-law-distributed ejection velo cities at h = 1000 m, u = 1650 m sec sec
-1 -1

and v0 = 10 m sec
-1

-1

(top right); and at h = 50 m, u = 100 m

and v0 = 20 m sec

(bottom right). The simulations with uniformly-

distributed ejection speeds were run with N = 106 particles while those for power-law-distributed ejection speeds utilised N = 107 particles.

16


5. Three-dimensional particle motion If grain motion is allowed in all three dimensions, the ejection velo city vector vL need not be vertical but at an angle z to the surface normal (Fig. 7). The (constant) horizontal component of the velo city vH = vL sin z propagates the particle at an azimuth to the (arbitrary) reference direction. We adopt the notation vN for the grain velo city component normal to the surface to distinguish it from vL . Laboratory experiments to-date provide evidence for preferential ejection of material at a narrow range of zenith angles (Gault et al., 1963; Koschny and Grun, 2001) (see also discussion in Kruger et al., ¨ ¨ 2000) so we adopt here a single value of z as a simplifying yet physically realistic assumption. Under this new notation, the ejection speed laws described by Eqs. 11 and 16 remain formally correct. The ensemble altitude and speed distributions derived in the previous Sections incorporate the pdf for what is now vN . The expressions given for the altitude (Section 3) remain valid if the parameters v0 , v+ and v+ are replaced with their respective pro jections onto the surface
+ - normal, that is v0N = v0 cos z , vN = v+ cos z and vN = v- cos z . In what

follows, we provide speed distributions for the case of power-law distributed ejection velo cities (Eq. 16). To derive the pdf of the speed v given the altitude h, one considers that there is now an additional random variable, the launch azimuth (see Fig. 7), uniformly distributed in the interval [0, 2) and independent of vL and tL . One can use this fact to write the joint probability density of , v and h as P ( = (, h, v), VL = vL (, h, v), TL = tL (, h, v))||J ||ddvdh = 17


1 P (VL = vL , TL = tL )ddtL dvL 2

(3 0 )

where J is the Jacobian of the transformation. It can be shown that the joint pdf of vL and tL is equal to the pdf for the 1-D case divided by cos z . The
- determinant of the Jacobian is equal to (v/vN ) vL 1 . Note that this expression - evaluates to vL 1 for z = 0 ie one recovers the result for vertical ejection as

intuitively expected. To arrive at the desired pdf, one integrates with respect to and makes use of Eq. 22. Since the integrand is independent of , the integration is trivial and the result is P (V = v|H = h) = g ( - 1 ) - 1 - v vL 2 cos z 0
-2

(h, v)

v 1 vN (h, v) fP (h; v0N )

(3 1 )

2 2 where vL = v2 + 2g h, vN = v2 cos2 z - (2g h) sin2 z and fP is given by Eq. 19.

Fig. 8 shows this pdf at an altitude of 100m superimposed on the results of numerical simulations as z increases from 1o (top left) to 10o (top right), 30o (bottom left) and finally 80o (bottom right). The adopted values of the mo del parameters are v0 = 10 m sec-1 , = 1.2. For reference, the case of vertical ejection for the same parameter values is represented by the second plot down from the top right of Fig. 4. The cutoff at low speeds is due to the constraint that vN 0 and corresponds to a mo de at v = tan z 2g h. As a general comment on the speed distributions in this paper, it is tempting in the absence of knowledge to the contrary and if experimentally-observed counts are relatively low as is the case in Fig. 8 - to intuitively expect the speed statistics to follow a different distribution eg. a maxwellian. This, however, would not be correct, at least under the assumptions of our mo del. In the final part of the paper, the potential implications of lifting some of those assumptions are discussed. 18


For a platform moving horizontally at speed u, can be measured from the direction of motion as Fig. 7 shows. In this reference frame, the speed w of the particle relative to the platform may be written as w2 = u2 + v2 - 2u 2g h + v2 sin z cos . (3 2 )

To derive the probability of w given h, we consider the transformation (, v) (, w) for which ||J || = w 1 - u sin z cos / v 2g h + v
2 -1

(3 3 )

The sought-for pdf is then
2

P (W = w|H = h) =

0

P (V = v (w, ) , = , H = h) ||J ||d fP (h; v0N )

(3 4 )

where the notation used in the integrand is from Eqs. 30 and 31. Note that it is necessary to solve Eq. 32 for v. This is a bi-quadratic which, in the first instance, admits to the real ro ots
2 v1, 2

=

u2 2 sin2 z cos2 - 1 + w ±2u sin z cos

2

w2 + 2g h + u2 sin2 z cos2 - 1

(3 5 )

if w2 u2 1 - sin2 z cos2 - 2g h. (3 6 )

Positive ro ots lead to solutions for v (hereafter referred to as "solutions" to distinguish them from the ro ots of Eq. 32) and define the subsets of the interval [0, 2 ] - as functions of w - over which the integration in takes place in Eq. 34. If, for given values of and w (say, and w ) only one of the two solutions exists, the integral of Eq. 34 is evaluated for that solution 19


only. If both solutions exist, the conditional probability density evaluates to P (v (w , ) , | h) = P (v1 , | h) + P (v2 , | h). A useful quantity for computational purposes is the pro duct of the two ro ots of Eq. 32, equal to w2 - u2
2

- 8u2 g h sin2 z cos2 .

(3 7 )

To illustrate the partition of (w, ) space in terms of the existence and the number of real solutions for v in Eq. 32 we have colour-co ded the different domains accordingly in Fig. 9 where z = 30 , u = 1650 m sec
-1

and h = 30

km. In the black region, the bi-quadratic has no real, positive solutions for v that satisfy Eq. 32. In the yellow region, exactly one such solution exists, namely v1 (Eq. 35). Finally, in the red region, both v1 and v2 are distinct, real and positive solutions of Eq. 32. The mo del probability distribution is illustrated in the top panel of Fig. 10 against the statistics of 107 Monte Carlo variates launched with velo cities distributed according to Eq. 16 with v0 = 100 m sec
-1

and = 1.2. The integral

in Eq. 34 has been evaluated numerically using the NIntegrate subroutine within the Mathematica package (Wolfram Research Inc., 2010). The main features of the observed distribution are the cutoff in the probability density below 1400 m sec
-1

and the two peaks at 1470 m sec

-1

and 1830 m sec

-1

re-

spectively. These are related to the boundaries between the different regions of (w, ) space identified in Fig. 9. By experimenting with different values of h, v0 and , we find that these are generic features of the distribution of the impact speed of ejecta on a moving platform in orbit and at altitudes of 10100 km above the surface. The two peaks approach each other as z decreases (compare the top and bottom panels of Fig. 9, the latter showing the same distribution but for z = 10 ). They merge into one as z vanishes, reverting 20


back to the expressions for vertical ejection (Fig. 6). Finally, we comment on the "dent" that appears in the distribution for z = 30 at w 1580 m sec-1 . It is not a real feature of the distribution, but arises due to the difficulty in evaluating the integral in Eq. 34 over the thinning part of the red domain as w u and /2 (Fig. 9). 6. Size-dependent ejection speed In the preceding sections we have considered ejection speed distributions which do not depend on ejecta size. On the basis of past work, it is reasonable to expect a dependence of the ejection speed on the sizes of both the ejectum and the impactor (Melosh, 1984; Miljkovi´ et al., 2012). In that case, the c ejection speed probability will be given by
s
I ,max

s s

E ,max

P (VL = vL ) =
s
I ,min E ,min

P (VL = vL , SE = sE , SI = sI ) dsE dsI (38)

where sE and sI denote the sizes (diameters) of the ejectum and impactor respectively. The integrand can be expressed in terms of the probability distributions of these two quantities through the chain rule for conditional probability: P (VL = vL , SE = sE , SI = sI ) = P (VL = vL |SE = sE , SI = sI ) P ( SE = s E | SI = s I ) P ( SI = s I ) . ( 3 9 ) To demonstrate the use of these expressions, we provide an example below for particular choices of the different distributions. For the impactor size range of interest, the last term on the right-hand-side can be approximated by a power law (Grun et al., 1985) ¨ P ( SI = s I ) ( s I / s
I ,min

)

-

/ sI , s

I ,min

< sI < s

I ,max

(4 0 )

21


if s

- I ,min

s

- I ,max

. In the recent study of the ejecta clouds of the jovian

mo ons Europa and Ganymede by Miljkovi´ et al. (2012) the distribution of c ejecta sizes was taken to be a deterministic function of ejection speed. In a probabilistic framework, this corresponds to the degenerate pdf for the first term on the right-hand side of Eq. 39: P (VL = vL |SE = sE , SI = sI ) = 1 if vL = C (sE /sI )-k and 0 otherwise (41) where the constant C depends on the target surface properties. For the ejecta size distribution, we adopt a power law P ( SE = s E | SI = s I ) = ( s E / s with (40), we require that s
- E ,min E ,min

)

-

/ sE , s

E ,min

< sE < s

E ,max

(4 2 )

where, for simplicity, we have assumed independence on impactor size. As s
- E ,max

. The exponents , k and are

all assumed to be positive. Upon integrating Eq. 38 we find that P (VL = vL ) vL s ++1 .
( +1)/k

(4 3 )

where the constant of proportionality (say K ) is K if s
-(+ +1) I ,min -( +1) E , m i n sI , m i n

C

-( +1)/k

.

(4 4 )

s

-(+ +1) I ,max

Comparing Eq. 43 with Eqs 16 and 19, we conclude that, for this particular parametric description of the ejection pro cess, the profiles of ejecta number density with altitude should follow a power law with an exponent of
+1 2k

. As this is a positive number by definition, the result predicts that the

number density of ejecta will increase with altitude. Therefore, the functional form of the distributions of the ejecta kinematics appears to be sensitivelydependent on the particular ejection mo dels adopted. 22


7. Conclusions and Discussion 7.1. Main Findings In this paper a metho dology has been described for deriving explicit steady-state probability distributions of the kinematic properties of impactgenerated dust in the vicinity of a planetary surface. This metho dology has been applied to the altitude and speed distributions of ejecta and validated against numerical simulations. Below we summarise the main findings: · the altitude and speed distributions of ejecta in a stationary frame and for power-law-distributed ejection speeds admit to analytical expressions that can be readily evaluated for arbitrary values of the defining parameters. · for power-law distributed ejection speeds, the number density of ejecta decreases with altitude as a power law and with an exponent that directly depends on the corresponding exponent of the ejection speed distribution. The scale height of the ejecta distribution, rather than being constant, increases with altitude. · close to the surface, the altitude distribution of ejecta exhibits a cusp that translates into a sheet containing the highest number density of ejected grains. The altitude and thickness of this sheet are functions of the parameters that describe the ejection speed law. Instantaneous grain speeds are power-law distributed with an exponent that, as is the case for the altitude distribution, directly depends on the corresponding exponent of the distribution of ejection speed.

23


· the power within the speed distribution of near-surface ejecta is concentrated at low - but non-zero - speeds. For vertical ejection, this distribution becomes flatter as altitude increases, eventually allowing the possibility of particles with vanishing speed. If ejection o ccurs at an angle to the vertical, the shape of the distribution remains qualitatively the same but a zero ejecta speed is no longer possible at any altitude. · for vertical particle ejection, the distribution of ejecta speed relative to a moving platform is qualitatively similar to that for the inertial case. If ejection o ccurs at an angle to the vertical, the distribution becomes qualitatively different with two separate maxima that move further apart as the zenith angle of ejection increases. Although probabilistic in nature, the collection of mo dels in this paper encapsulates most of the information necessary to predict absolute quantities such as the particle number density n(h) and the flux F (h). A conceptuallevel algorithm to arrive at these quantities is as follows: n(h) is the pro duct of the altitude pdf (Section 3) with the number of particles in flight at a given time (assumed constant for a steady state pro cess; Eq. 9). For a powerlaw speed distribution this is given by Eq. 17 where the rate parameter (number of particles per unit time) can be estimated via quantitative mo dels of ejecta pro duction for a given impactor flux (e.g. Krivov, 1994). The particle flux F (h) - although dependent on the orientation of the incident surface (McDonnell et al., 2001) - may be evaluated as n(v, h)vdv. The

first term in the integrand represents the number of particles with speed v and at height h; it can be expressed as n(h)p(v|h) where p(v|h) is the 24


conditional pdf of the speed given the height from Section 4. 7.2. Implications for in situ measurements The above can be viewed as predictions for specific features to be sought in the measured properties of impact-ejected dust near airless planetary bo dies. At the same time, the class of mo dels intro duced here are, by their very nature, suitable for treating planetary dust exospheres as natural laboratories of the fundamental pro cesses of dust mobilisation and transport in the solar system. For example, ejection speed law parameters as inferred from measurements can be compared to the results of laboratory experiments and hydro co de simulations. The knowledge gained can be applied to the lowgravity regime relevant to surface pro cesses on NEOs and small bo dies in general. In addition, although the mo dels have been constructed through the frequentist approach to probability theory, they can also be utilised by bayesian inference techniques to extract information such as the most likely launch lo cations and speeds of grains at the surface. This would be particularly useful in exploring the dependence between the properties of the impactors and those of their ejecta (see also point on steady state assumption below). Recent dust measurements offer a suitable proving ground for our statistical mo del and an opportunity to pursue some of the above ob jectives, wholly or in part. The Lunar Dust Experiment (LDEX) impact ionisation dust detector operated from October 2013 until April 2014 onboard the Lunar Atmosphere and Dust Environment Explorer (LADEE) spacecraft in orbit around the Mo on (Delory, 2014). The charge collected by the instrument allows the grain mass and speed to be estimated from laboratory-derived 25


calibration curves (James and Szalay, 2014). Our finding of a power-law dependence of the dust distribution (Eq. 19) implies that LDEX dust counts, when binned in altitude and corrected for the different residence time of the spacecraft in each altitude bin, will also follow a power law. Unlike the case far from the bo dy however (Krivov et al., 2003), at the low altitudes where LDEX operated the exponent of this power law directly depends on the assumed mo del of the ejection physics, specifically the exponent of the ejection speed distribution (Eq. 16). It follows that this exponent can be measured directly from the LADEE dust counts and that any pro cess that mo difies it will result in a different power-law fit to the measurements. Furthermore, in Section 5 it was found that adoption of a single, non-zero value for the grain ejection zenith angle leads to a double-peaked profile for the distribution of grain impact speed relative to a moving platform (Eq. 34 and Fig. 10). Therefore - and assuming that grain impact speeds can be measured with a precision of 50 m sec
-1

or better - the existence of two

peaks in LDEX data would provide evidence that a particular ejection angle dominates for real impacts. Measurements best suited for this purpose would be those collected near the pericentre and apo centre of LADEE's orbit; at those lo cations the radial component of the velo city vector vanishes, in line with our assumption of a horizontally-moving platform. 7.3. Caveats To place our findings in the proper context it is necessary to highlight here several important assumptions that were made in the course of this study. The following is not intended as an exhaustive list. Probably the most important is that fragment ejection speed and size 26


were assumed to be uncorrelated. Since our probability distributions concern the relative number of ejecta independently of their size, the measurements against which they will be compared will be dominated by a relatively narrow range of sizes. Nevertheless, understanding the consequences of intro ducing a dependence between size and ejection speed is important, not only to gauge how the results of this paper apply to the real solar system but also to allow the exploitation of additional information, such as size, momentum and kinetic energy in both existing and future datasets. A first step in this direction has been made in the penultimate Section of the paper where a metho d to treat the dependency between ejecta size and speed is presented. To demonstrate it, we worked through an example for a particular functional description of the ejection pro cess. We find a power-law dependence of the ejecta number density on altitude similar to that found in Section 3 but with a positive exponent that is a function of the parameters describing both the size and speed distribution of ejecta. This emphasises the point made earlier that the distributions of the ejecta kinematics are sensitive to the ejection physics and bo des well for constraining the latter through measurements by orbiters at low altitudes or by landers at the surface. A full probabilistic treatment of this class of problems using the same metho dology is outside the scope of the present work, but we note that it can (a) encompass mo dels where surface grains are mobilised by electrostatic forces (Stubbs et al., 2006; Hartzell et al., 2013), and (b) yield conditional distributions involving the physical properties of the dust (e.g. distribution of grain size/mass at a certain altitude or moving at a certain speed), also relevant to the measuring capabilities of current and future instrumentation

27


(Hirai et al., 2010; Sternovsky et al., 2010; Carpenter et al., 2012). In particular, it should allow one to test the efficiency of electrostatic vs ballistic mobilisation of dust as competing mechanisms for the pro duction of dust exospheres. Next, the assumption of a steady state requires that the ensemble properties of the dust population within an altitude bin are time-invariant. In practice, the assumption holds if the variation of the measured statistical quantity over time is significantly smaller than the measurement itself. It is not immediately clear that this is true, since it depends on the efficiency of the source pro cess (impact flux and number of ejected grains pro duced per unit time). It is, however, possible to emulate such a state by averaging measurements over time and for as long as the source pro cess do es not vary. For the lunar case, and given the non-isotropic background meteoroid flux in the 0.1-10 mm size range (Campbell-Brown and Jones, 2006; Campbell-Brown, 2007), one may expect source variations as the surface normal to a given lo cation on the surface scans through the full range of angles with respect to the Earth's apex every month. Shorter-term variations in the impactor flux are also expected, manifesting themselves in the Earth's atmosphere as meteor showers and outbursts (Jenniskens, 1994, 1995). These should be taken into account for dust cloud mo delling as they add to the value of in situ measurements in understanding the ejection pro cess for different impactor populations. Finally, high-speed impacts by the primary ejecta population (Zo ok et al., 1984; Grun et al., 1985) will pro duce secondary ejecta which have not been ¨ taken into account here. If important, their low kinetic energy relative to

28


that of primary ejecta renders them more likely to mo dify the properties of the ejecta cloud at the low end of the range of altitudes considered here. The probabilistic framework in which the present mo del has been developed should allow treatment of multiple generations of ejecta. It would be interesting to determine, for example, if the near-surface features found in Section 3 persist under these circumstances. This will be explored in future work. Acknowledgements Astronomical research at the Armagh Observatory is funded by the Northern Ireland Department of Culture, Arts and Leisure (DCAL). The author wishes to thank Dr Mih´ly Hor´nyi and an anonymous reviewer for their a a comments which improved the paper considerably; and Dr David Asher, for numerous discussions during the course of this research that prevented wasting time on less-than-pro ductive avenues of investigation and for his comments on a draft version of the manuscript. References Berg, O. E., Wolf, H., Rhee, J. W., 1976. Lunar soil movement registered by the Apollo 17 cosmic dust experiment. In: Interplanetary Dust and Zo diacal Light. Springer-Verlag, Berlin, pp. 233­237. Campbell-Brown, M., 2007. The meteoroid environment: shower and sporadic meteors. ESA Special Publication 643, 11­21. Campbell-Brown, M., Jones, J., 2006. Annual variation of sporadic radar meteor rates. Mon. Not. Royal Astron. So c. 367, 709­716. 29


Carpenter, J. D., Fisackerly, R., De Rosa, D., Houdou, B., 2012. Scientific preparation for lunar exploration with the European Lunar Lander. Planetary and Space Science 74, 208­223. Criswell, D. R., 1973. Horizon-glow and the motion of lunar dust. In: Photon and Particle Interactions with Surfaces in Space. D. Reidel Publishing Co., Dordrecht, Holland, pp. 545­556. Delory, G., 2014. Lunar Atmosphere and Dust Environment Explorer PDS Mission Description. Tech. rep., Ames Research Center, National Aeronautics and Space Administration. Elphic, R. C., Delory, G. T., Grayzeck, E. J., Colaprete, T., Hor´nyi, M., a Mahaffy, P., Hine, B., Boroson, D., Salute, J. S., 2011. The science behind NASAs Lunar Atmosphere and Dust Environment Explorer. In: Annual Meeting of the Lunar Exploration Analysis Group. Vol. 13. p. 2056. Gault, D. E., Sho emaker, E. M., Mo ore, H. J., 1963. Spay ejected from the lunar surface by meteoroid impact. NASA Technical Note TN-D 1767. Grun, E., Fechtig, H., Zo ok, H. A., Giese, R. H., 1985. Collisional balance of ¨ the meteoritic complex. Icarus 62, 244­272. Hartzell, C. M., Wang, X., Scheeres, D. J., Hor´nyi, M., 2013. Experimental a demonstration of the role of cohesion in electrostatic dust lofting. Geophysical Research Letters 40, 1038­1042. Hirai, T., Sasaki, S., 8 co-authors, 2010. Lunar dust monitor for the orbiter of the next Japanese lunar mission SELENE-2. In: Geophys. Res. Abs. Vol. 12. p. 14014. 30


James, D., Szalay, J., 2014. Lunar Atmosphere and Dust Environment Explorer (LDEX PDS Software Interface Specification). Tech. rep., Ames Research Center, National Aeronautics and Space Administration. Jenniskens, P., 1994. Meteor stream activity I. The annual streams. Astronomy & Astrophysics 287, 990­1013. Jenniskens, P., 1995. Meteor stream activity. II: Meteor outbursts. Astronomy & Astrophysics 295, 306­335. Koschny, D., Grun, E., 2001. Impacts into ice-silicate mixtures: Ejecta mass ¨ and size sistributions. Icarus 154, 402­411. Krivov, A. V., 1994. On the dust belts of Mars. Astronomy & Astrophysics 291, 657­663. Krivov, A. V., Sremevic, M., Spahn, F., Dikarev, V. V., Kholc shevnikov, K. V., 2003. Impact-generated clouds around planetary satellites: spherically-symmetric case. Planetary and Space Science 51, 251­ 269. Kruger, H., Krivov, A. V., Grun, E., 2000. A dust cloud of Ganymede main¨ ¨ tained by hypervelo city impacts of interplanetary micrometeoroids. Planetary and Space Science 48, 1457­1471. Kruger, H., Krivov, A. V., Sremevic, M., Grun, E., 2003. Impact-generated ¨ c ¨ dust clouds surrounding the Galilean mo ons. Icarus 164, 170­187.

31


McCoy, J. E., 1976. Photometric studies of light scattering above the lunar terminator from Apollo solar corona photography. In: Pro c. Lunar Planet. Sci. Conf. Vol. 7. pp. 1087­1172. McCoy, J. E., Criswell, D. R., 1974. Evidence for a high altitude distribution of lunar dust. In: Pro c. Lunar Planet. Sci. Conf. Vol. 5. pp. 2991­3004. McDonnell, T., McBride, N., Green, S. F., Ratcliff, P. R., Gardner, D. J., Griffiths, A. D., 2001. Near Earth Environment. In: Grun, E., Gustafson, ¨ B. A. S., Dermott, S. F., Fechtig, H. (Eds.), Interplanetary Dust. Astronomy and Astrophysics Library. Springer, Heidelberg, pp. 163­231. Melosh, H. J., 1984. Impact ejection, spallation, and the origin of meteorites. Icarus 59, 234­260. Miljkovi´, K., Hillier, J. K., Mason, N. J., Zarnecki, J. C., 2012. Mo del of dust c around Europa and Ganymede. Planetary and Space Science 70, 20­27. Rennilson, J. J., Criswell, D. E., 1974. Surveyor observations of lunar horizon glow. The Mo on 10, 121­142. Rhee, J. W., Berg, O. E., Wolf, H., 1977. Electrostatic dust transport and Apollo 17 LEAM experiment. In: COSPAR Space Research. Vol. 17. Pergamon, New York, pp. 627­629. Severny, A. B., Terez, E. I., Zvereva, A. M., 1975. The measurements of sky brightness on Lunokho d-2. The Mo on 14, 123­128.

32


Sremevic, M., Krivov, A. V., Spahn, F., 2003. Impact-generated clouds c around planetary satellites: asymmetry effects. Planetary and Space Science 51, 251­269. Sternovsky, Z., Hor´nyi, M., Grun, E., Srama, R., 2010. The Lunar Dust a ¨ Experiment (LDEX) instrument on board the Lunar Atmosphere and Dust Environment Explorer (LADEE) mission. In: Cospar Scientific Assembly. Vol. 38. p. 478. Stubbs, T. J., Vondrak, R. R., Farrell, W. M., 2006. A dynamic fountain mo del for lunar dust. Advances in Space Research 37, 59­66. Wolfram Research Inc., 2010. Mathematica Version 8. Wolfram Research, Inc., Champaign, Illinois. Zo ok, H. A., Lange, G., Grun, E., Fechtig, H., 1984. Lunar primary and ¨ secondary micro craters and the micrometeoroid flux. In: Proc. Lunar Planet. Sci. Conf. Vol. 15. pp. 965­966. Zo ok, H. A., McCoy, J. E., 1991. Large scale lunar horizon glow and a high altitude lunar dust exosphere. Geophysical Research Letters 18, 2117­2120.

33


List of Figures 1 Mo del (red dashed curves) and simulated (bars) distributions for the probability density function (top) and cumulative distribution function (bottom) of altitude for a population of particles launched with the same speed. . . . . . . . . . . . . . . 36 2 Mo del (bold curves) and simulated (gray histogram) distributions of altitude for uniformly distributed (left panels) and power-law distributed (right panels) ejection speeds. . . . . . . 37 3 Top: Altitude probability density functions for a power-law speed distribution and different values of the ejection speed limit v0 with a lighter colour corresponding to a higher value. Bottom: Effective inverse scale height for the distribution with v0 = 1 m sec-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Mo del (bold curves) and simulated (gray histogram) distributions of v given h for a uniformly distributed (left panels) and power-law distributed (right panels) ejection speeds. . . . . . . 39 5 Distribution of v given h for a power-law distribution of ejection speeds for five different values of the altitude h where the speed scale is linear (top) and for four different values of the altitude on a logarithmic speed scale (bottom). A lighter colour corresponds to a higher altitude. . . . . . . . . . . . . . 40 6 Distribution of the speed of ejecta w at altitude h relative to a moving platform for a uniform (left panels) and a power-law (right panels) distribution of ejection speeds. . . . . . . . . . . 41

34


7

Graphical representation of the kinematics of an ejected particle relative to the surface and to a moving platform as implemented in our mo del. See main text for notation. . . . . . . 42

8

Distribution of the speed v of ejecta at altitude h in the threedimensional problem for a power-law distribution of ejection speeds and for different values of the zenith angle z of ejection. 43

9

Partition of (w, ) space in terms of the number of solutions for the speed v of the particle in Eq. 32 for z = 30 . . . . . . . 44

10

Distribution of the speed of ejecta w relative to a platform moving horizontally for a power-law distribution of ejection speeds in the three-dimensional problem and for z = 30 (top) and z = 10 (bottom). . . . . . . . . . . . . . . . . . . . . . . 45

35


0.20 0.15

PH h

0.10 0.05 0.00 0 2000 4000 6000 8000 10 000

Height m
1.0 0.8

PH h

0.6 0.4 0.2 0.0 0 2000 4000 6000 8000 10 000

Height m
Figure 1:

36


0.00007 0.00006 0.00005 v
L

Uniform Launch Speed 490 m s 1 , v
L

0.0005 0.0004

500 m s

1

Power Law Launch Speed v0 1 m s 1 , 1.2

PH h

PH h
0 20 000 40 000 60 000 80 000

0.00004 0.00003 0.00002 0.00001 0

0.0003 0.0002 0.0001 0.0000 0 500 1000 1500 2000

Height m
0.00003 v 0.000025 0.00002 0.000015 0.00001 5. 10
6 L

Height m
0.005 0.004 Power Law Launch Speed v0 10 m s 1 , 1.2

Uniform Launch Speed 300 m s 1 , vL 500 m s 1

PH h

PH h
0 20 000 40 000 60 000 80 000

0.003 0.002 0.001 0.000 0 500 1000 1500 2000

0

Height m
0.00007 0.00006 0.00005 v
L

Height m
0.0005 0.0004

Uniform Launch Speed 10 m s 1 , v
L

500 m s

1

Power Law Launch Speed v0 50 m s 1 , 1.2

PH h

PH h
0 20 000 40 000 60 000 80 000

0.00004 0.00003 0.00002 0.00001 0

0.0003 0.0002 0.0001 0.0000 0 500 1000 1500 2000

Height m

Height m

Figure 2:

37


1

0.01

PH h

10

4

10

6

10

8

10

4

0.001

0.01

0.1

1

10

100

Height km
1000

P 1 km dP dh

100 10 1 0.1 0.01 0.001

0.01

0.1

1

10

100

Height km
Figure 3:

38


0.020 v
L

Uniform Launch Speed 30 m s 1 , vL 200 m s 1 Height 10m

0.25 0.20 0.15 0.10 0.05 0.00

Power Law Launch Speed v0 10 m s 1 , Height 1.2 10m

PV v H h

0.015

0.010

0.005

0.000 0 100 200 300 400 500

PV v H h

0

20

40

60

80

100

Speed m s
0.020 v
L

Speed m s
0.08
1

Uniform Launch Speed 30 m s 1 , vL 200 m s Height 100m

PV v H h

PV v H h

0.015

0.06

Power Law Launch Speed v0 10 m s 1 , 1.2 Height 100m

0.010

0.04

0.005

0.02

0.000 0 100 200 300 400 500

0.00 0 20 40 60 80 100

Speed m s
0.020 v
L

Speed m s
0.030 0.025 Power Law Launch Speed v0 10 m s 1 , 1.2 Height 1km

Uniform Launch Speed

PV v H h

0.015

PV v H h
300

30 m s 1 , vL 200 m s 1 Height 1km

0.020 0.015 0.010 0.005

0.010

0.005

0.000 0 50 100 150 200 250

0.000 0 20 40 60 80 100

Speed m s
0.020 v
L

Speed m s
0.010 Power Law Launch Speed v0 10 m s 1 , 1.2 Height 10km

Uniform Launch Speed

PV v H h

0.015

PV v H h
300

30 m s 1 , vL 200 m s 1 Height 10km

0.008 0.006 0.004 0.002 0.000

0.010

0.005

0.000 0 50 100 150 200 250

0

50

100

150

200

250

300

Speed m s

Speed m s

Figure 4:

39


1 0.1

PV v H h

0.01 0.001 10 10 10
4

5

6

0

100

200

300

400

500

Speed m s
1

0.01

PV v H h

10

4

10

6

10

8

1

5

10

50

100

500

Speed m s
Figure 5:

40


1.0 v
L

Uniform Launch Speed

0.5

Power Law Launch Speed v0 10 m s 1 , Height 1.2 1km

0.6 0.4 0.2 0.0 1650

PW w H h

PW w H h

0.8

30 m s 1 , vL 200 m s 1 Height 1km

0.4 0.3 0.2 0.1 0.0 1650

1652

1654

1656

1658

1660

1662

1652

1654

1656

1658

1660

Speed m s
0.08 v
L

Speed m s
0.6 0.5 Power Law Launch Speed v0 20 m s 1 , Height 1.2 50m

Uniform Launch Speed

PW w H h

0.06

PW w H h
220

30 m s 1 , vL 200 m s 1 Height 100m

0.4 0.3 0.2 0.1

0.04

0.02

0.00 100 120 140 160 180 200

0.0 100 102 104 106 108 110 112 114

Speed m s

Speed m s

Figure 6:

41


-u w u
v
H

z

v

Figure 7:

L

42

v h

N

v


0.08

Power Law Launch Speed v0 10 m s 1 , 1.2 Height 100m z 1 deg

0.08

Power Law Launch Speed v0 10 m s 1 , 1.2 Height 100m z 10 deg

PV v H h

0.06

PV v H h
100

0.06

0.04

0.04

0.02

0.02

0.00 0 20 40 60 80

0.00 0 20 40 60 80 100

Speed m s
0.08 Power Law Launch Speed v0 10 m s 1 , 1.2 Height 100m z 0.04 30 deg 0.08

Speed m s
Power Law Launch Speed v0 10 m s 1 , 1.2 Height 100m z 0.04 80 deg

PV v H h

0.06

PV v H h
100

0.06

0.02

0.02

0.00 0 20 40 60 80

0.00 0 50 100 150 200

Speed m s

Speed m s

Figure 8:

43


150

Azimuth deg

100

50

0 1200

1400

1600

1800

2000

w ms
Figure 9:

44


0.005

PW w H h

0.004 0.003 0.002 0.001 0.000 1400 1600 1800 2000

Speed m s
0.015

PW w H h

0.010

0.005

0.000 1400 1600 1800 2000

Speed m s
Figure 10:

45