The azimuthal distribution of ejecta mass from oblique impacts into sand
Alice C. Quillen Sean Doran




a1]Alice C. Quillencor1
alice.quillen@rochester.edu

a1]Sean Doran
sdoran3@u.rochester.edu
















[a1]Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA







[cor1]Corresponding author


We measure ejecta mass as a function of azimuthal and impact angle for 104 m/s oblique impacts into sand.  We find that the ejecta mass distribution is strongly sensitive to azimuthal angle with up to 8 times more mass in ejecta on the downrange side compared to the uprange side. Crater radii,  measured from the site of impact, are measured at different impact and azimuthal angles.  Crater ejecta scaling laws are modified to depend on azimuthal and impact angle. We find that crater radii are sensitive to both impact and azimuthal angle but the ejecta mass as a function of both angles can be estimated from the cube of the crater radius without an additional angular dependent function.  The ejecta distributions are relevant for processes that depend upon the integrated properties of intermediate velocity impacts occurring in the outer solar system and possibly during planetesimal formation. 
 
 
 









The azimuthal distribution of ejecta mass from oblique impacts into sand
    [
    April 29, 2024
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1





§ INTRODUCTION


Because ejecta from an impact can escape an asteroid or planetesimal, the velocity and mass distributions of ejecta affect the total momentum imparted to the asteroid and as a consequence, its deflected trajectory <cit.>.    
The ratio of the change in center of mass momentum and the projectile momentum is known as 
the momentum transfer efficiency parameter, and is relevant for mitigation strategies for deflection of potentially hazardous asteroids <cit.>.  
Over a long period of time, impacts cumulatively exert a torque on a spinning body, which can reduce its rotation rate <cit.>.  
The distribution of ejecta in a lower velocity impact regime (10 to 50 m/s) is relevant for estimating 
collision evolution of asteroids <cit.>, accretion, erosion and particle transport on young and forming planetesimals <cit.>. 







While most craters on asteroid and satellite surfaces, including the Moon, are nearly round, impacts on astronomical bodies rarely have projectile velocity vector nearly normal to the surface. 
We take the impact angle θ_I to be the angle between the plane tangent to the surface at the point of impact and the projectile momentum vector, as seen in a reference frame in which the surface is stationary.  With this convention, grazing impacts have low impact angle. 
More than 50% of impacts are expected to occur at impact angles between 30 and 60^∘ <cit.>. Normal impacts on a homogeneous and isotropic level substrate exhibit 
rotational symmetry about an axis normal to the surface that intersects the site of impact. Azimuthal symmetry facilitates integrating over the ejecta mass and velocity distributions,  consequently scaling laws based on dimensionless parameters for estimating crater size and ejecta mass and velocity distributions predominantly apply to normal impacts <cit.>.


The momentum transfer caused by an impact can be computed from the integral of the fraction of ejecta distribution that escapes the planetesimal (e.g., ).

An estimate for the torque caused by impacts on asteroids requires taking into account the distribution of impact angles in the population of projectiles <cit.>. 
This motivates us to improve upon estimates for the ejecta distributions of oblique impacts. 
The low or intermediate velocity regime (about 20 to 100 m/s) would be relevant for impacts from particles in the protosolar nebula <cit.> and for impacts in the outer solar system  
<cit.>.



Impact craters formed at normal impact angle obey scaling relationships for crater size and volume 
<cit.> 
based on the dimensionless parameters  

    π_2     =  g_a a_pj/ u_pj^2
    π_3     = Y_a/ρ_a u_pj^2
    π_4     = ρ_a/ρ_pj .

Here g_a is the gravitational acceleration at the site of impact,
ρ_a is the density of the substrate, Y_a is substrate strength, u_pj is projectile velocity, ρ_pj is projectile density, and a_pj is the projectile radius. 
Recent compilations of crater dimensions suggest that these scaling laws are remarkably good at matching crater properties, such as radius and volume, over a wide range of impact and substrate properties, and including a low velocity regime <cit.>. 
Assuming that a projectile acts like a point source when considering crater-related phenomena,  <cit.> developed power-law scaling relations for the properties of ejecta curtains, with 
 ejecta properties such as ejecta speed and ejected mass  dependent on the horizontal distance from the site of impact. 

Ejecta mass and velocity distribution functions for oblique impacts, when integrated,  are directly related to the momentum transfer efficiency and the change in spin caused by an impact.  
The ejecta distributions derived in the point source approximation by <cit.> were modified to depend upon azimuthal angle and ejecta angle and then fit to velocity and mass ejecta distributions measured from shock physics code simulations <cit.>.  
 

<cit.> used a power-law form for the ejecta distributions, based on impact experiments by <cit.>, to estimate a cumulative torque on asteroids due to impacts. Improved measurements of   ejecta distributions for oblique impact would help improve estimates for momentum transfer and torque caused by impacts.

Ejecta from high velocity oblique impacts have been tracked using high speed cameras <cit.>
giving measurements of both ejecta angle and velocity distributions as a function both impact angle θ_I and azimuthal angle ζ measured in the substrate plane.  
Azimuthal variations in ejecta angle and velocity imply that there is asymmetry in the subsurface response  <cit.> and 
this was verified for oblique impacts into sand using embedded accelerometers <cit.>.  
Simulations of high velocity impacts have been used to measure the sensitivity of ejecta velocity, angle and mass distributions to both impact and azimuthal angle <cit.>. 


Ejecta particle velocity can be tracked in videos taken with high speed cameras <cit.>, but because particles in an ejecta curtain can obscure material behind them, it is more challenging to measure the ejecta mass distribution from 
videos. 
Trays laid on the surface or mounted above the surface can be used to capture ejecta which  
 later are weighed, giving constraints on the mass distribution of the ejecta curtain  <cit.>.  
We build upon our prior experimental study of intermediate velocity oblique impacts
 <cit.> by similarly using  trays to catch ejecta at different azimuthal angles. 
 

Crater depth profiles at different impact angles are 
used to model the azimuthal  and impact angular dependence of crater radius. 
We extend ejecta crater scaling models for normal impacts 
to describe the azimuthal and impact angle dependence of the ejecta mass distribution.   
Our distributions may be used in models that integrate over ejecta properties to estimate momentum 
transfer and torque from intermediate velocity oblique impacts. 





§ EXPERIMENTS




Our experiments build upon those described by <cit.>, with  
projectile and substrate properties summarized in Table <ref>.   
Illustrations of our experiments are shown in  Figure <ref>. 


Small plastic spherical projectiles (referred to as pellets or BBs) are launched with an airsoft gun at a speed of v_imp = 103 to 105 m/s. 
The substrate is fine sand with grain semi-major axis mean value a_s ≈ 0.3 mm, as described in previous experiments  <cit.>.
Dimensionless constants π_2, π_3, π_4, used to characterize crater scaling regimes 
are also listed in Table <ref> along with their definitions (following ).  
Experimental measurements suggest that rocky granular or regolith systems would have an effective strength of order Y_a ≈ 500 Pa <cit.> and we adopt that value to compute  
the dimensionless parameter π_3. 











Following <cit.>, a substrate particle is considered ejected as it crosses the plane that was the 
surface of the substrate prior to impact, as shown in Figure <ref>a.  In this plane, 
the distance from impact is x and v_ej is the ejection velocity. After the transient crater has formed, we take crater radius R_cr to be the radius from the site of impact where the crater surface crosses the level of the surface prior to impact, and as shown in Figure <ref>a.   The radius R_rim is the 
distance from impact site to rim peak, also measured in the substrate level plane. 
For oblique impacts we use the same definitions but these two radii are functions of 
impact angle and azimuthal angle; R_rim(θ_I,ζ) and R_cr(θ_I,ζ).  

The crater volume V_cr is the volume of the crater that lies below the  surface level prior to impact. 
For normal impacts, we use a subscript n, giving crater radius R_cr,n, 
crater rim radius R_rim,n, 
and crater volume V_cr,n.   Crater radii and volume for normal impacts 
along with dimensionless parameters  π_R, π_V computed from these quantities are also 
listed in Table <ref>. 

Crater scaling laws predict that a normal impact in the strength regime has crater radius 

    R_cr,n(π_3,π_4) = a_pj(4π/3)^1/3 H_2 
    π_3^-μ/2π_4^-ν,
 and 
in the gravity regime

    R_cr,n(π_2,π_4)  = a_pj(4π/3)^1/3 H_1  π_2^-μ/2+μπ_4^-6ν/2+μ ,

following Table 1 by <cit.>, but with n subscripts to denote a normal impact. 
The parameters H_1,H_2 and exponents μ,ν are dimensionless but can depend upon the material properties  <cit.>. 

A crater is in the strength regime if  π_3^1+ μ/2π_4^ν/π_2 > 1  <cit.>. 
With exponents μ,ν∼ 0.4 typical of granular systems  <cit.> and Y_a = 500 Pa, 
 our experiments have π_3^1.2π_4^0.4/π_2 ∼ 1,  similar to unity.  Our experiments lie near the line dividing strength and gravity regimes.   Because we work in a granular system and 
 its effective strength has not been measured, we often assume that we are in the gravity regime, following 
 
<cit.> who suggested that impacts into granular systems would always be in the gravity regime.  
However, <cit.> argued that even a low level of strength would put smaller and lower velocity impactors in a low gravity environment in the strength regime.  

Setting equation <ref> or <ref> equal to our normal impact crater radius and using values for the dimensionless parameters from Table <ref>, 
we estimate H_1 ≈ 0.6 in the gravity regime or H_2 ≈ 0.7 in the strength
regime.   These coefficients are similar to those estimated 
for different experiments (see  Table 3 by ).

We describe oblique impacts with impact angle θ_I which is the angle between the projectile velocity vector and the substrate plane.  The impact angle is zero if the impact is grazing and is 
π/2 if the impact is normal to the surface, as shown in Figure <ref>a.    
The azimuthal angle ζ is measured 
using the site of impact as origin and in the plane perpendicular to the surface normal n̂ as shown in Figure 
<ref>b.  We adopt the convention that ζ = 0 in the uprange direction (opposite the d̂ vector), and toward the projectile launcher, following <cit.>. 




 §.§ Ejecta mass at different azimuthal angles









We carried out a set of oblique impact experiments at different impact angles specifically to measure ejecta mass. 
To collect ejecta from our laboratory impacts, we constructed a set of eight nearly identical paper trays to catch ejecta.  When placed next to each other, they cover an octagonal area around the impact site, as shown in Figure <ref> and in the photo in Figure <ref>.  Paper was used to keep the trays light, reducing compression of the sand substrate from the weight of the tray. Each paper tray weighs about 5.25 g.  
A circular area 8 cm in diameter, approximately matching crater diameter, was cut out from the apexes of the trays to allow ejecta to escape the impact region.  Each tray covers an azimuthal angle of dζ = 45^∘, as illustrated in Figure <ref>.  
Each tray is labelled by its central azimuthal angle.   

More ejecta mass originates at lower velocity near the crater rim than at higher velocity 
near the crater center (e.g., ). 
If a tray overlaps the crater rim, it would block the ejecta curtain.  
To capture the most ejecta mass, we designed the trays
so that the tray inner edges are close to the crater rim, but do not cover it.  
 In the mid-velocity regime of our 104 m/s impacts, craters are nearly round, 
with larger major to minor axis ratio at lower impact angles 
(at θ_I = 20^∘, the major to minor crater axis ratio is only 1.18; see Table 5 by ).
As a consequence, we cut the tray inner edges so that they are arcs of a circle.   The three straight
sides of the trays have folded edges, as shown on the lower right in Figure <ref> so that ejecta
does not bounce out just after impact or fall out when we move the tray to weigh it. 
The inner central curved tray edge is flat so that ejecta is not blocked by the tray.  
We aim the projectile so that the resulting crater is centered with respect to the central 
region of sand that is not covered by the trays, as illustrated in Figure <ref>.
We did not use smaller trays because the ejecta mass collected in each tray is low, 
often less than a gram.  

To improve projectile aim, 
we rebuilt the airsoft gun mount and digital servo trigger system so that it is held 
more securely than in our previous experiments.  The airsoft gun is fired electronically 
without touching it.  
We also mounted an adjustable laser sight to an extension of the airsoft gun's picatinny rail which 
helps us aim the projectile and measure the impact angle. 
Prior to carrying out experiments with the trays, we do a set of impacts to adjust the airsoft gun tripod angle and location so that the crater is centered
in the washtub container.  
The impact angle is measured with the laser sight and a large protractor that is drawn 
on a sheet of poster board, and is accurate to approximately 2 degrees. 


Prior to each impact experiment, the surface is raked with a wire rake with wire tongs that are about 10 cm deep,  as described in our prior experiments <cit.>.  After raking, 
 the surface is leveled with a rectangular metal bar so that its level is equal to the height of the rim of the tub holding the sand. Afterward raking and leveling, the empty paper trays are placed on the surface 
with positions and orientations illustrated in Figure <ref>. 


After an impact we examine the crater.  A few experiments were discarded because the crater rim 
did not lie within the central hole made by the trays.  Otherwise,  
after each impact, the weight of the sand collected in each tray is measured with a digital scale. 

We discuss uncertainties in our ejecta mass measurements. 
The distance from impact site to the center of the tray pattern 
gives a variation in the azimuthal angle (measured with origin at the impact site) subtended 
by each tray.   
Despite making improvements in our trigger servo system, we found that the site of impact 
is not exactly at the same position every time we fire a projectile.  By examining 
crater locations with respect to a laser target, we find that the difference between desired
and actual impact site has a standard deviation of about a cm. 
Ejecta scaling is done with origin at the impact site <cit.>, as illustrated in Figure <ref>a, however, we attempted to center our trays 
with respect to the crater center, as illustrated in Figure <ref>.   
Craters from oblique impacts are not centered about the site of impact.  
To characterize the distance between 
crater center and impact site,  <cit.> measured 
 the distance d_ai between the midpoint of uprange and downrange rim peaks and the site of impact.  The distance d_ai  is a maximum of 1.49 cm
at impact angle θ_I = 40^∘, and below 1.1 cm for the remaining impact angles, 
 as listed in Table 5 by <cit.>.     The crater at θ_I=40^∘ may be particularly elongated
in part because the projectile passed through the ejecta curtain. 

We consider the value  d_ai = 1.1 cm (a value characteristic of our experiments for θ_I between 30 and 60^∘), 
the angle subtended by the ζ = 180^∘ downrange tray at its inner edge is about 0.8 times dζ = 45^∘ and the angle subtended by the 
ζ = 0 uprange tray is 1.4 times dζ = 45^∘.   This implies that our ejecta masses
are lower than they should be
 in the downrange trays and higher than they should be in the uprange trays. 
A similar size variation in angle subtended by the trays is caused when the impact site
is 1 cm closer to the ζ=90^∘ tray than the 270^∘ tray, in which case
the trays on the closer side would  
receive more ejecta than the opposite side.   
The worst case error in subtended angle of about 40% seems large,  however we find that 
the azimuthal variations in 
ejecta mass are so large that they dwarf uncertainties caused by errors 
in impact site with respect to the trays. 





Because craters are smaller and more elongated at grazing angles, with θ≤ 30^∘ the trays would miss 
ejecta near the crater rim, particularly along the crater minor axis at ζ∼ 90^∘ and 270^∘. 
However, variations in ejecta mass caused by crater elongation are much smaller than those caused by craters
that are not centered with respect to the trays. 
High velocity ejecta can land past the outer edge of our trays. 
At an impact angle θ_I = 40^∘ we measured the fast ejecta mass that escaped  our trays altogether, 
finding that it is at most 0.16 g and negligible. 

We carried out 6 experiments at each of 7 impact angles ranging from 20 to 80^∘ with an increment of 10^∘.  The average ejecta masses in g (of 6 measurements) as a function impact angle and tray central azimuthal angle are listed in Table <ref>.   
Each panel in Figure <ref> shows the ejecta masses in grams as a function 
of azimuthal angle ζ (along the horizontal axis) at a different 
impact angle.  In Figure <ref>,  the dotted lines show individual measurements from each experiment.
The large dots at each azimuthal angle show the average ejecta masses computed from 6 experiments.  
The vertical error bars  show the standard deviations computed
from the scatter about the average values and are also listed in Table <ref>.  
The horizontal error bars in Figure <ref> show the d ζ = 45^∘ angle subtended by each tray.  

Figure <ref> shows that ejecta mass from oblique impacts depends on azimuthal angle,  
with more mass ejected in the downrange direction than in the uprange direction.  
At θ_I=50^∘,
there is a dip in the ejecta mass  at azimuthal angle ζ = 180^∘, 
in the down range direction.  At this impact angle, the projectile ricochets at a low
angle, grazing the crater rim (see Figure 4 by ).  
The projectile itself could have blocked part of the ejecta curtain near ζ = 180^∘, 
reducing the ejecta mass at this azimuthal angle. 

Figure <ref> shows that there is scatter in our ejecta mass measurements which can primarily
be attributed to variations in impact location with respect to the tray pattern. 
Scatter can also be  
attributed to density variations within the sand (though we mitigated by raking the sand prior to impact),  surface level variations of the sand prior to 
impact due to uneven leveling and disturbance caused by tray placement (less than about 1 mm), and relative variations (of order 0.3 cm) in tray locations and orientation (a few degrees) between experiments.   

We did not carry out experiments of ejecta mass at the grazing angle of θ_I = 10^∘ because 
the crater is more elongated and smaller than at higher impact angles, 
and because it was more difficult to accurately aim the projectile at low impact angles.   



Despite the scatter in individual measurements and the low mass of ejecta measured 
in each tray (often well below 1 g per bin), Figure <ref> shows that azimuthal variations in ejecta mass 
are robustly measured in our experiments. 
The ratio between the ejecta mass in the downrange bin to that in the uprange bin 
is large, ranging from about 3 at θ_I = 80^∘ to about 8 at θ_I = 50^∘
(see Figure <ref> below).
The differences between up and downrange ejecta are so large that they exceed the errors  
introduced by centering the trays at the crater center rather than the impact site and by variations 
in impact location. 



 





 §.§ Total ejecta mass as a function of impact angle


Figure <ref> shows total ejecta mass at each impact angle, θ_I.  These are computed by
summing the ejecta mass from the azimuthal bins.   Black x's show total ejecta mass for each
experiment and the large blue triangles show the averages computed from experiments at each impact angle. 
The vertical error bars are estimated from the scatter in the total ejecta mass measurements.
The horizontal error bars represent uncertainty in impact angle. 
The ejecta mass measurements are in g and have axis on the left.
Crater volume measurements are shown with orange circles, are in cm^3,  and have axis on the right. 
Crater volume measurements are 
taken from Table 5 by <cit.> except that at θ_I = 80^∘ which we corrected after reexamining the crater profile. 


Figure <ref> shows that the total mass in ejecta is approximately proportional to crater volume, 
 as expected from prior studies <cit.>.  
The low ejecta mass at grazing impact angles (lower than 50^∘) is related to the reduction in crater volume that was previously noted by <cit.> and that was attributed to energy carried away via ricochet of the projectile.  A crater mass can be estimated 
from the product of crater volume times substrate density; M_cr = ρ_a V_cr. Crater mass is
 the mass that was originally located inside the crater. 
 In Figure <ref> we have adjusted the y-axis scales so that
crater volume measurements lie on top of crater ejecta mass measurements, giving 
 ratio of ejecta to crater mass 

    M_ej(θ_I)/M_cr(θ_I)≈ 0.35 .
 

The shock physics code impact simulations by <cit.> predict a ratio of 
ejecta to crater mass of 0.37 for their high velocity impacts 
(see their Figure 15) with a lower value of 0.25 for their lowest impact velocity (1 km/s) simulation. 
Our value of 0.35 is consistent with this prediction, despite the difference in impact regime (our impact 
velocity is at least an order of magnitude lower).  In hypervelocity experiments into layered sand    <cit.> attributed the relatively large value of crater mass 
compared to ejecta mass  to 
compactification of the substrate and deformation associated with radial expansion and rim uplift (which 
 exceeds ejecta curtain depth ). 
Our low value of ejecta to crater mass does not necessarily imply that our experiments 
fail to capture most of the ejecta, even though our trays do not capture some low velocity material 
 ejected very near the crater rim. 







§ CRATER RADII AS A FUNCTION OF AZIMUTHAL ANGLE



Models for the ejecta velocity and mass distribution can relate the properties of the ejecta curtain 
to the crater radius <cit.>.   However, for oblique impacts, the crater radius 
depends on both impact and azimuthal angles. To relate our measurements of the azimuthal ejecta 
mass distribution to models for the ejecta curtain we first measure the crater radius 
as a function of impact and azimuthal angles. 

<cit.> measured crater depth profiles for oblique impacts by taking a video 
while slowly stepping a line-laser across the crater.  
The videos were processed to measure crater depth as a function of position at 8 different impact angles and
creating eight two dimensional crater depth arrays <cit.>.    
Here we analyze these eight crater depth profiles again but  
to measure crater and rim radius as a function of azimuthal and impact angles. 

 <cit.> measured crater depth profiles from 8 experiments, at impact angles
 ranging from  θ_I = 10^∘ to 80^∘ in increments of 10^∘. 
For each of the eight computed two dimensional crater depth arrays,   
we extracted the depth as a function of distance from the site of impact at different azimuthal angles.  Positions along a desired angle ζ were extracted using linear interpolation from the 2d-arrays depth arrays.  The depth profiles at each azimuthal angle were smoothed using a 1 dimensional Savinsky-Golay filter
of window length 3 mm and polynomial order 3.  
The maximum height along the smoothed profile gave the radius of the rim peak R_rim(θ_I,ζ).   Within the rim we identified the radius which was at the same depth as the undisturbed surface prior to impact, R_cr(θ_I,ζ).  


The resulting rim and level crossing radii are plotted in Figure <ref> as a function of azimuthal angle 
for each of 8 impact angles θ_I. 

The rim diameter and crater diameter for a normal impact listed in Table <ref> we computed
from the medians of the rim radii and the medians of the crater radii 
 at different azimuthal angles using the depth profile from the experiment
 at an impact angle of θ_I= 80^∘. 
 Twice the rim radius differs slightly from the rim to rim crater diameter along the major
 axis of 7.1 cm previously listed in Table 5 by <cit.>.   
We give an updated value for crater volume in Table <ref> for a normal impact
based on the same depth profile 
as we suspect that there was a 
minor error in the value reported in Table 5 by <cit.>. 













The size of azimuthal variations in ejecta mass exceed those in crater radius.  To show this, 
we examine the ratio of ejecta mass in the downrange direction to that in the uprange 
direction and we compare this ratio to the ratio of the downrange crater radius
to uprange crater radius. 
in cigure <ref> at each impact angle,  black dots show the ratio of the maximum 
ejecta mass to the minimum ejecta mass using the average values plotted in Figure 
<ref>.  The maximum values tend to be downrange and the minimum values
uprange.   At each impact angle,  with red squares,  we show the ratio of the maximum to 
minimum crater radii that are also shown in Figure <ref>.  
Figure <ref> shows that the ejecta mass amplitudes of azimuthal 
variation are higher than the amplitudes of azimuthal variation in the crater radius. 

Figure <ref> shows that the uprange/downrange asymmetry in ejecta mass is large. 
Due to the placement of our ejecta trays (centered at crater center rather than impact point), the 
asymmetry could be even larger.  
If we correct by the extreme error values estimated in section <ref>),  the  ratio of maximum to minimum ejecta mass could increase by a factor as large as 1.4/0.8 = 1.7.   







 §.§ Fitting the crater radius as a function of impact and azimuthal angles
 




Using least-square minimization we fit a function 
 to the radii R_cr(θ_I,ζ) and R_rim(θ_I,ζ) 
 that we measured from the crater depth profiles.
The function we choose extends the function for crater radius adopted 
 by <cit.> that is linear in impact angle θ_I and depends on cosζ. 
We use the function 
 
    R(θ_I,ζ) =  a_00 + ∑_j=1^2 ∑_k=0^2  a_jk( θ_I- π/2)^j   (cosζ)^k

with coefficients a_00 and a_jk with integers j>0 and k≥ 0. 
 For the fit to R_cr, the coefficient a_00=R_cr,n for a normal impact. 
We found that a polynomial function in θ_I required fewer coefficients for the same quality of
fit (as measured from the reduced χ^2) than one that only contained trigonometric functions of θ_I. 
A least squares fitting routine in python ( with the Trust Region Reflective algorithm) was used to find the coefficients a_ij that minimize χ^2.  Uncertainties 
 for each coefficient are 1 σ errors that are estimated from the diagonals of the covariance matrix
 and using the variance of the residuals as an estimate for the standard deviation of each data point.  
The best fitting coefficients (via equation <ref>) for our measurements 
of R_cr and R_rim (shown in Figure <ref>) are listed 
 in Table <ref> along with the standard deviation of the residuals.  
 in Figure <ref> the best fitting functions are shown in the top panels with measurements 
 in the middle panels and residuals  (data subtracted
 by model) shown in the bottom panels.  
 Figure <ref>a  shows our fit to R_cr measurements, and 
 Figure <ref>b shows our fit to R_rim measurements in both cases fit varying  all
 6 coefficients in equation <ref>.  
 
In Figure <ref> the colored dots show the crater radii (R_cr(θ_I,ζ)) we measured
as seen from above the surface. Here the +x axis is in the uprange direction. 
Dot size and color depend
upon impact angle.   In Figure <ref>a we plot the best fitting function (with coefficients 
for equation <ref> and coefficients in the column labelled R_cr in Table <ref> 
for each impact angle 
with lines, and with line color matching the dots at the same impact angle.  







Figure <ref>a and Figure <ref>a show that our fitting function (equation <ref>) matches
our measurements for the crater radius fairly well.  
The largest residual errors are at impact angle θ_I = 40^∘.  The elongation of the crater
along the downrange direction  at this impact angle 
can in part be attributed to the ricochet of the projectile 
that grazed the crater rim along the negative x axis downrange of impact <cit.>.  

<cit.> fit the crater radii of their impact simulations with 
a function (their equation 9)  similar to equation <ref>, dependent upon azimuthal and impact angle,  
but only containing the two coefficients   
a_00 = R_cr,n and  a_11 = - R_cr,n1/21/100180/π = -0.29R_cr,n.






In Figure <ref>c we show a least-square model that minimizes  
a version of the function in equation <ref> containing only two free parameters, 
a_00 and a_11.  The resulting best fit two coefficient model and 
residuals are shown in Figure <ref>c.  The values 
 of the two coefficients for this fit are listed in the rightmost column in Table <ref>. 
The coefficients of our best fit two coefficient model has a ratio of a_11/a_00 = -0.24 which is similar to the value of -0.29 of the model by <cit.> (their equation 9). 
The residuals between the two-component model and our measurements are higher for the two coefficient model than that with six coefficients,   as would be expected since
 fewer free parameters are varied in the χ^2 minimization.   Residuals between 
 our measurements and the two-component model are primarily high at low impact angle
where our crater radii are smaller than at higher impact angle.    Figure <ref>b,  
where we show the crater shapes predicted by the two-component model (as dashed lines) along with our crater shapes (shown with dots), also shows that our measured crater radii are smaller than predicted by the two-component model at low impact angle. 

The impact simulations by <cit.> were of a 7 km/s impact.   
The function used by <cit.> does not have a smaller crater at low impact angles,
which was true of our experiments.  Craters are smaller  because 
 impact energy and momentum was carried away by the projectile
during ricochet at grazing impact angles in our lower velocity experiments <cit.>.  
The simulations by <cit.> were of a high velocity (7 km/s) impact which would have been more
likely to melt or vaporize the projectile than cause it to ricochet.  
The two component function fit by <cit.> to their simulated impacts for crater radius
as a function of azimuthal and impact angle was close to the radii we measured (particularly at impact angles
greater than 40^∘) despite the difference in impact regime. 



§ CRATER EJECTA DISTRIBUTION MODELS



The ejecta model for normal impacts by <cit.> assumes that the velocity and mass distributions for most of the ejecta are independent of projectile radius.   
This assumption is described as the “point source approximation”.  
Ejecta velocity and mass 
 is a power law function of x, the distance between ejecta launch site and impact site. 
The power law functions are approximately truncated at an  
inner radius that is approximately equal to the projectile radius a_pj and outer radius 
that is approximately equal to the crater radius  R_cr, 

as  illustrated in Figure 9.



The ejecta scaling model by <cit.> has total mass ejected at all positions up to x 
from the site of impact 

    M_ej,n(<x) = k_n ρ_a( x^3  - n_1 a_pj^3)    for   n_1 a_pj < x< n_2 R_cr,n.

This power law form for the ejecta mass distribution is illustrated in Figure <ref>.  
The parameter k_n is dimensionless but can depend upon the material properties <cit.>.  <cit.> found that parameters n_1 ≈ 1.2 and n_2 ≈ 1. 


Following <cit.>, we modify the coefficients used in the crater ejecta scaling 
laws by <cit.> to become functions of impact and azimuthal angles θ_I,ζ.  
We modify equation <ref> to consider the ejecta mass in an azimuthal bin of size d ζ in radians 

    dM_ej(<x,θ_I, ζ) dζ   =ρ_a/2 π k(θ_I,ζ) ( x^3  - n_1 a_pj^3) d ζ
           for    n_1 a_pj <  x<n_2 R_cr(θ_I, ζ).

The factor of 2 π arises because the total ejecta mass is found by integrating
over the azimuthal angle ζ.   The function k(θ_I,ζ)  
has limit 

    lim_θ_I →π/2 k(θ_I,ζ)   = k_n

where k_n is the coefficient for a normal impact in equation <ref>. 
The crater radius is dependent upon azimuthal and impact angles and 

    lim_θ_I →π/2 R_cr(θ_I,ζ)  = R_cr,n

where R_cr,n is the crater radius for a normal impact in equation <ref>. 
We assume that coefficients n_1 and n_2 are independent of 
impact and azimuthal angle. 

The total ejecta mass (integrated out to the crater radius) in an azimuthal bin of width dζ
in radians is 

    dM_ej (θ_I, ζ) ≈ρ_a/2 π k(θ_I,ζ) R_cr^3(θ_I,ζ)  d ζ

where we have used n_2 ≈ 1 found by <cit.> in equation <ref> for the
maximum value of ejecta radius and assumed that crater radius is much larger than the projectile radius; 
R_cr,n≫ a_pj.  

For the function k(θ_I,ζ), we adopt a form for the function;  

    k(θ_I,ζ) = k_n  + k_1 cosθ_I cosζ .

Using equation <ref> and parameters based on our model for R_cr(θ_I,ζ) 
(described in section <ref>) 
we fit k_n and k_1 to the mean ejecta mass  
we measured as a function of
impact and azimuthal angle.  The averages at each impact angle
consistent of mean values from measurements 
of 6 experiments per impact angle and at each of 8 possible azimuthal bins. 
Our least-squares minimizing fitting procedure is the same as 
described in section <ref>. The model that minimizes χ^2 is shown in 
Figure <ref>.  
Best fitting parameters for the function in equation <ref> are 

    k_n    = 0.19 ± 0.01   g
    
    k_1    = 0.02 ± 0.02    g

with a standard deviation of 0.28 g in the residuals (model subtracted by measurements)
that are shown in the bottom panel in Figure <ref>.   

Our best fitting value for k_n  is at the lower end of the range 
0.2 to 0.5 for normal impacts into 
substrate materials inferred by <cit.> 
via comparison of their ejecta scaling model to experimental measurements.  
Our trays do not catch all the lowest velocity ejecta, which may in part account
for our low k_n value. 




Our best fitting value for k_1 is near zero,  which implies that the function k(θ_I,ζ) is only weakly dependent upon impact and azimuthal angles. Variations in the total ejecta mass (integrated in ejection launch position x) as a function of azimuthal angle in equation <ref> are primarily due to the sensitivity of the crater radius to impact and azimuthal angles.  
 
The function used by  <cit.> to describe azimuthal variations in the ejecta mass, 
 k(θ_I,ζ) = k_n/2πexp[-0.02cos(ζ)cos(θ_I)]
(their equation 8, which we have modified to be per bin in ζ).  This function is 
 weakly dependent upon impact and azimuthal angles. 
Our measurements of the ejecta mass azimuthal mass distribution 
confirm this weak dependence.   We suspect that Figure 12c by <cit.> is probably not
showing their function k(θ_I, ζ) (given in their equation 8) 
as the plotted function is more sensitive to azimuthal and impact angles than predicted by
their function.   Figures 12a, b and d by <cit.> may  
have flipped or mislabelled x-axes as the plotted functions appear inconsistent with their equations 6, 7 and 9.  
Our experimental measurements are in agreement with two of the functions proposed by <cit.>;  
their equation 8 (for k(θ_I,ζ)) and their equation 12 (for R_cr(θ_I,ζ)), despite some confusion caused by their Figure 12. 











§ DISCUSSION AND SUMMARY


In this study we have measured the azimuthal ejecta mass distribution for intermediate (∼ 104 m/s) velocity 
oblique impacts into sand.   Our measurements extend our characterization of the intermediate velocity 
regime for oblique impacts in a granular system, building upon the experimental study by <cit.>. 
Using paper trays to catch ejecta, 
we have measured the azimuthal ejecta mass distribution as a function of impact angle.
Even though craters are nearly round, 
the ejecta curtains are strongly asymmetrical, with more ejecta launched in the downrange than
uprange direction, confirming the study by <cit.>.   
The ratio of ejecta mass in the downrange bin to the uprange one is large and ranges from about 3 
at an impact angle of θ_I = 80^∘ to at least 8 at an impact angle of θ_I = 40^∘. 

The large ejecta mass in the downrange direction is consistent with the strong, high   
amplitude subsurface pulses on the downrange side measured with embedded accelerometers by <cit.>.   

Crater rims from oblique impacts are not centered about the site of impact. 
Using crater depth profiles measured by <cit.> we have measured 
the crater radius R_cr(θ_I,ζ) (a distance from
the site of impact illustrated in Figure <ref>a),  
as a function of impact and azimuthal angles.   
We find that the azimuthal ejecta mass distribution is approximately proportional to the cube of crater radius 
R_cr(θ_I,ζ).   This implies that the function k(θ_I,ζ) 
(defined in equation <ref>),  is 
approximately constant. The insensitivity of the 
function k(θ_I,ζ) describing the ejecta mass distribution 
 to impact and azimuthal angles  would facilitate 
computing quantities associated with the cumulative affects of impacts on small bodies, 
including angular momentum drain <cit.> 
and erosion <cit.>.   

The shape of subsurface pulses measured by <cit.> from accelerometers at the particular subsurface locations  
were remarkably similar for oblique impacts at different impact angles (see their Figure 17). 
<cit.> suggested that it might be possible to describe 
the subsurface pulses and the resulting ejecta velocity field by rescaling the amplitude of the velocity field.  
Our finding, that the ejecta mass azimuthal distribution 
depends primarily on crater radius, supports this proposal.  There may be a way to characterize 
 ejecta distribution and impact generated subsurface motions via an angular and time dependent function 
 that also gives R_cr(ζ,θ).  
 
<cit.> modified ejecta scaling laws to describe simulated oblique impacts 
 with four angular dependent functions.  
We have experimentally characterized two of these functions:  crater radius R_cr(θ_I,ζ) and the function k(θ_I,ζ) (defined in equation <ref>), 
though in a lower velocity regime and for a granular substrate, sand.  
Our angular dependent functions for crater radius and 
k(θ_I,ζ) describing the ejecta mass distribution are remarkably similar to those presented by
<cit.>.  

We found that the azimuthal asymmetry in ejecta mass was so large that we could detect it  
with unsophisticated impact experiments.   The biggest source of error in our experiments is likely 
is due to variations in projectile aim.  Impact experiments with more accurate targeting could confirm  
and improve upon our measurements and study the ejecta mass distribution from 
oblique impacts at different impact velocities. 
In future work, we also hope to experimentally characterize 
two additional functions of impact and azimuthal angles, θ_I and ζ, for the coefficient C_1 and exponent μ used by <cit.> to describe the ejecta velocity as a function of ejecta launch position. 
We would also like to characterize 
functions describing angular variations in ejecta angle as ejecta angle is also sensitive 
to azimuthal and impact angles and ejecta launch position <cit.>. 



Acknowledgements: We thank Bingcheng Suo for helping us set up the experiments. 
This work has been supported by NASA grant 80NSSC21K0143. 

Data availability:  
The data that support the findings of this study and code used to generate all figures in this manuscript are openly available at <https://github.com/aquillen/azimuthal_mass>.  

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