Model of CabeusA1 and LCROSS ejecta plume
canopus56@yahoo.com 9-21-2009

Model results for crater, sunlight shadow mask and ejecta plume

Characteristics

Cabeus A -
Selenographic coordinates : 34.15 W 81.04 S  
Diameter : ~19 km
Depth : Not stated
At impact on Oct. 9, 11 : 30 UT, Sun angle : 8.27 alt 331.33 az 
Ejecta curtain -
Truncated cone or inverted lampshade : 5 km dia.bottom, 5 km high, 10 km dia.top.
Edge angle : 45 degree in baseline.  35 degrees in alternative scenario.

Crater depth to diameter relationships - typical and Cabeus A1

Hale and Grieve (1982) crater scaling equation for 17 km to

51 kmH=0.000589 × (D^1.969)

where H = crater depth and D is crater diameter.

Plot[0.000589 × (x^1.969), {x, 0, 17}]

0.000589 (17^1.969)

But Cabeus A1 is shown on NASA LCROSS color coded topography maps as being on the order of 1 km to 1.5 km deep.   The sun angle the crater at impact is 8.27 degrees.

Plot[Tan[-8.27 Degree]× x, {x,0,9.5}]

Tan[-8.27 Degree]× 4.0

The LCROSS Team’ s Crater Selection Presentation (ver.9 - 15), states the plume will have to rise 630 meters reach sunlight.  With a sun angle of 8.27 degrees at the time of impact and assuming the impact occurs 5 km off center, the sunlit plane is 580 meters below the rim. This also suggests a crater depth of approximately 1 to 1.5 kilometers.

Modelling the crater bowl

The above discussion suggests a parabola 1.2 km deep and a radius of 9.5 km for satellite crater Cabeus A1.

(((x - 0 ))^2) = 1.2 × 62.5 × (y - 0)

y = (x^2)/(1.2/62.5)

Plot[(x^2)/(1.2*62.5),{x,0, 9.5}]

(9.5^2)/(1.2*62.5)

And in 3 D form, the crater bowl would be :

RevolutionPlot3D[(x^2)/(1.2*62.5),{x,0, 9.5},Mesh -> None, PlotStyle->Directive[Opacity[1.0],Gray], MeshShading->{Gray,Automatic} ,BoxRatios->{1,1,0.05} ]

Modeling the sun plane

The plane of the Sun' s light is representated by a line angled at - 8.27 degrees.

Tan[8.27 Degree]

Plot[Tan[8.27 Degree]× x-0.180825,{x,-9.5,9.5}]

- rendered as a plane :

Plot3D[Tan[8.27 Degree]× x-0.180825,{x,-9.5,9.5},{y,-9.5,9.5},Mesh->None,PlotStyle->Directive[Yellow,Specularity[White,20],Opacity[0.4]],BoxRatios->{1,1,0.05} ]

Modeling the ejecta plume

The plume is a lampshade figure 5 km at the bottom and 10 km at top.   The bottom radius is 2.5 km; the top 5km.  A line between points 2.5, 0 and 5,5 models the edge of the plume.  This implies an edge angle for a right triangle with sides of 2.5, 5.0 and a line with a slope of 2.0.

The bright edge of the ejecta plume can be modeled using a line equation function:

Plot[2×(x  -2.5), {x,2.5,5.0}]

Line line is  rotated in 3 D to create the curtain model:

RevolutionPlot3D[2×(x-2.5), {x,2.5, 5.0},Mesh -> None, PlotStyle->Directive[Opacity[0.5],Red], MeshShading->{Gray,Automatic} , BoxRatios->{1,1,0.5} ]

Combining the crater, sun plane and ejecta curtain

The following combines the crater and the sunline, but not the ejecta curtain.

Show[
RevolutionPlot3D[(x^2)/(1.2*62.5),{x,-0, 9.5},Mesh -> None, PlotStyle->Directive[Opacity[1.0],Gray], MeshShading->{Gray,Automatic},BoxRatios->{1,1,0.05} ],
Plot3D[Tan[8.27 Degree]× x-0.180825,{x,0,9.5},{y,-9.5,9.5},Mesh->None,PlotStyle->Directive[Yellow,Specularity[White,20],Opacity[0.4]],BoxRatios->{1,1,0.05}]
]

All three images are combined, but the translation is wrong on the curtain.  All three elements will not scale properly within a single Mathematica rendering box.  The curtain is truncated.

Show[
RevolutionPlot3D[(x^2)/(1.2*62.5),{x,-0, 9.5},Mesh -> None, PlotStyle->Directive[Opacity[1.0],Gray], MeshShading->{Gray,Automatic},BoxRatios->{1,1,0.05} ],
Plot3D[Tan[8.27 Degree]× x-0.180825,{x,0,9.5},{y,-9.5,9.5},Mesh->None,PlotStyle->Directive[Yellow,Specularity[White,20],Opacity[0.4]],BoxRatios->{1,1,0.05}],
RevolutionPlot3D[2×(x-2.5), {x,2.5, 5.0},Mesh -> None, PlotStyle->Directive[Opacity[0.5],Red], MeshShading->{Gray,Automatic} , BoxRatios->{1,1,0.5} ]
]