C---+----1----+----2----+----3----+----4----+----5----+----6----+----7-- PROGRAM ASTEROID !Every celestial body bound to the Solar System has by nature a !minimum and maximum velocity which is determined by total energy !(kinetic + gravitational). As a result, impact objects (e.g., !asteroids, comets, etc.) bound to the Solar System for every !planet has a "minimum" and "maximum" impact speed relative to !the gravitional and orbital characteristics of each planet. !This program calculates maximum and minimum planetary surface !impact speed of an impacting body (e.g., asteroids, comets, !etc.) bound to the Solar System in unperturbed co-planar !elliptical orbit. It can be shown that co-planar motion of the !impact body coincides with both maximum and minimum impact speed !for any Solar System planet under the condition of unperturbed !elliptical motion. ! !The largest and smallest possible impact speeds for any planet !occur when the impact body originates infinitely far away from !the Solar System with zero velocity and zero kinetic energy !(i.e, maximum total energy, and least bound condition for the !impact object). It follows from this program that minimum !impact speed for all gaseous planets (Jupiter, Saturn, Uranus, !Neptune) is less than planetary escape velocity. For all other !planets including Earth, minimum impact speed is greater than !escape velocity. ! !A measured impact speed greater than the maximum impact speed !listed in this program for any planet would be interpreted as !either an object originating from outside the Solar System or !an object undergoing close fly-by perturbation effects or !fragmentation effects such as which happened with comet !Shoemaker Levy 9 which impacted planet Jupiter in 1994. !Shoemaker Levy 9 was torn apart by differential gravitational !force from Jupiter and was separated into at least 21 detectable !fragments. The impact speed of the fragments was reported to be !around 55-58 km per second from various studies. This impact !speed is close to the planetary escape velocity of around 59.7 !km per second for Jupiter. External gravitational force from !Jupiter tore apart the comet in a previous encounter, inducing !an impact by reducing the comet center of mass kinetic energy !and altering its orbital path into a direct collision. ! !The analysis of impact speeds with planets involves potential !body breakup like Shoemaker Levy (i.e., comets which are loosely !bound "dirty snowballs" in space) and other effects including !orbital resonance between the planet and impacting body. An !asteroid or comet with a revolution period around the Sun at !twice the period of the planet in near encounter will experience !a resonance effect (like physically changing center of mass on a !swing to increase gravitational torque and height). In truth, !minimum impact speeds for all planets approach zero, and maximum !impact speeds may exceed many 100's of km per second for small !impact objects. This program provides only a baseline for !minimum and maximum collision speeds for impact bodies under the !simplest constraint of unperturbed elliptical co-planar orbits. ! !Impacts with the Earth have always occurred and will occur again !with devastating results. In July 2006 a large asteroid (XP 14) !estimated to be about a half mile (~800 m) wide in diameter !passed about 269,000 miles from the Earth (for comparison, the !mean distance between the Earth and Moon is about 238,000 !miles). In June 2002 a large space rock estimated to be ~30 m !wide missed the Earth by about 75,000 miles and was detected !only after it passed by. An asteroid named "Apophis", measuring !about a quarter mile in diameter was up until year 2006 thought !to be the biggest future threat of a collision with Earth in !year 2029. ! !The Smithsonian Astrophysical Observatory at Harvard University !currently lists about 780 "Potentially Hazardous Asteroids" !(PHA's) that in some distance future could be on a collision !path with the Earth. The last significant impact with the Earth !occurred in 1908 in Tunguska, Siberia which flattened trees over !an area more than 800 square miles and was caused by perhaps a !comet or asteroid fragment that disintegrated in the Earth's !atmosphere above the surface - the effects were similar to a !large nuclear explosion without the radiation fallout effect. !The size of that object has been estimated to be only about 60 !meters in diameter. ! !Meteor showers (often associated with comets in elliptical orbit !about the Sun) are well documented. Below is a table (Table !21.1 from "Exploration of the Universe" (fourth edition) by !George O. Abell). The table shows that all listed meteor impact !speeds with the Earth lie within minimum impact speed (13.7 !km/sec) and maximum impact speed (74.3 km/sec) as determined in !this program (see output text file "asteroid.out" generated from !this program). ! !Table 21.1 (G. O. Abell): ! ! Date of Impact Period ! Meteor maximum Velocity Associated of comet ! shower display (km/sec) comet (yr) !-------------------------------------------------------------- !Quadrantid Jan 3 43 --- 7.0 !Lyrid Apr 21 48 1861 I 415.0 !Eta Aquarid May 4 59 Halley 76.0 !Delta Aquarid Jul 30 43 --- 3.6 !Perseid Aug 11 61 1862 III 105.0 !Draconid Oct 9 24 Giacobini-Zinner 6.6 !Orionid Oct 20 66 Halley 76.0 !Taurid Oct 31 30 Encke 3.3 !Andromedid Nov 14 16 Biela 6.6 !Leonid Nov 16 72 1866 I 33.0 !Geminid Dec 13 37 --- 1.6 ! !Both minimum and maximum impact speeds are calculated and !tabulated in this program for all nine planets. Maximum and !minimum collision speeds for any planet occur at Perihelion !(closest position to Sun for a planet). This program lists !three tables. The first is for Perihelion, the second (for !comparison) is for Aphelion (farthest position point from Sun !for a planet). The third is a summary. The tables printed to !screen are also printed to an ASCII text file named !"asteroid.out". ! !This is a non-relativistic problem and impact speeds can be !determined by separating the problem into two velocity !components and then combining them. The first component is the !revolution speed of the planet around the Sun, and the other is !the speed of the comet (or asteroid, etc.) generated by the !gravitational energy of the Sun, impacted planet, and other !Solar System bodies. At impact (at the radius of the planet) !only the mass of the Sun (~745 times the combined mass of all !planets) and the impacted planet mass are important by dominant !balance. ! !Adding gravitational perturbations at impact radius from the !other planets, the Mars-Jupiter Asteroid Belt, Kuiper Belt !(beyond Neptune), and the outer Oort cloud does not alter impact !speeds for any impacted planet by more than around 0.1 km per !second. Nevertheless, this program includes the gravity-induced !perturbation speeds from the other non-impacted planets (an !input option for comparison of this effect). For simplicity, !this program does not adjust for Sun-planet center of mass in !any of the impact calculations for the planets. This affect on !impact speeds for any planet is small. As an example, adjusting !for the center of mass between the massive planet Jupiter and !the Sun alters revolution speed of Jupiter around the Sun by !only around 0.007 km per second. ! !This program tabulates impact speeds for both gravitational and !revolution components, and for both Perihelion and Aphelion !planet positions (first two tables). Maximum gravity-induced !object impact speed for any planet occurs at Perihelion when all !other planets are aligned co-linearly on the same side of the !Sun. In contrast, minumum gravity-induced impact speed occurs !at Aphelion when all other planets are co-linearly aligned on !the opposite side of the Sun. This program applies planetary !gravitational perturbations for maximum and minimum effects !when the planets are all aligned co-linearly. Despite the many !different scenarios, it turns out for all nine planets that both !maximum and minimum impact speeds for a comet or asteroid (etc.) !occur at Perihelion position where the revolution speed of the !impacted planet around the Sun is largest. ! !This program first computes maximum and minimum revolution !speeds of all planets around the Sun which occur at Perihelion !and Aphelion, respectively. It can be shown from conservation !of angular momentum that (a = semi-major axis, e = eccentricity, !G = gravitational constant, M = mass of Sun, r = distance !vector, v = velocity vector, "X" denotes cross product, and !the '|' symbols denote the magnitude of cross product) ! ! sqrt(G*M*a(1-e**2)) = |r X v|. ! !It can also be shown (definition of eccentricity) that the !distance at Perihelion is given by a*(1-e). (Similarly, the !distance at Aphelion is a*(1+e)). From these relationships it !follows that the velocity at Perihelion is given by !sqrt(G*M*(1+e)/(a(1-e))) = Vavg*sqrt((1+e)/(1-e)) where !Vavg is the time-averaged revolution speed of the planet around !the Sun (for Earth it is about 29.8 km/sec). The velocity at !Aphelion is sqrt(G*M*(1-e)/(a(1+e))) = Vavg*sqrt((1-e)/(1+e)). ! !For both Perihelion and Aphelion planet positions this program !tabulates planet name, semi-major axis (i.e., time-averaged !mean distance to the Sun), mean impact speed caused by !revolution of the planet around the Sun, impact speed caused !by gravity (Sun, impacted planet, other planets), minimum impact !speed when trajectory paths of the impact body and impacted !planet are in the same co-linear velocity direction, and maximum !impact speed when their trajectory paths are in opposite !co-linear velocity directions. ! !Author: Dr. Jerry R. Ziemke ! PARAMETER(EARTHMASS=5.977E24, SUNMASS=1.991E30) PARAMETER(EARTHRADIUS=6.37E6, AU=1.495E11, G=6.6743E-11) CHARACTER*9 PLANETNAME(9) REAL RADIUS(9),DISTANCE(9),MASS(9),VREV(9),VESC(9) REAL VREVP(9),VGRAVP(9),VREVA(9),VGRAVA(9),ECC(9) DATA DISTANCE /0.387, 0.723, 1.0, 1.524, 5.203, 9.539, 19.18, & 30.06, 39.53/ DATA RADIUS /0.382, 0.949, 1.0, 0.532, 11.19, 9.46, 4.01, & 3.88, 0.180/ DATA MASS /0.0552, 0.815, 1.0, 0.107, 318.0, 95.2, 14.5, & 17.1, 0.00216/ DATA ECC /0.206,0.007,0.017,0.093,0.048,0.056,0.047, & 0.009,0.248/ DATA PLANETNAME /'MERCURY','VENUS','EARTH','MARS','JUPITER', & 'SATURN','URANUS','NEPTUNE','PLUTO'/ !Calculate planetary average revolution speed around the Sun: DO I=1,9 VREV(I)=SQRT(G*SUNMASS/(AU*DISTANCE(I))) ENDDO !Adjust VREV average speed to Perihelion speed VREVP: DO I=1,9 VREVP(I)=VREV(I)*SQRT((1+ECC(I))/(1-ECC(I))) ENDDO !Adjust VREV average speed to Aphelion speed VREVA: DO I=1,9 VREVA(I)=VREV(I)*SQRT((1-ECC(I))/(1+ECC(I))) ENDDO !Calculate escape velocity for each planet: DO I=1,9 VESC(I)=SQRT(2.*G*MASS(I)*EARTHMASS/(RADIUS(I)*EARTHRADIUS)) ENDDO !Calculate gravity-induced surface incident speed at !Perihelion and Aphelion planetary positions: WRITE(*,*)'Choose (option):' WRITE(*,*)' 1) Include gravitational perturbations from the' WRITE(*,*)' other eight planets for each impacted planet' WRITE(*,*)' 2) DO NOT' READ(*,*) IPERT DO I=1,9 S1=0 IF (IPERT.EQ.1) THEN !Maximum (Perihelion) perturbations from other planets (all !other planets aligned co-linearly on same side of Sun): DO J=1,9 IF (J.NE.I) THEN S1=S1+MASS(J)*EARTHMASS/(ABS(DISTANCE(J)-DISTANCE(I))*AU) ENDIF ENDDO ENDIF !Add Sun + impact planet (+ other planets): VGRAVP(I)=SQRT(2*G*(SUNMASS/(DISTANCE(I)*AU*(1-ECC(I)))+ & MASS(I)*EARTHMASS/(RADIUS(I)*EARTHRADIUS) & +S1)) ENDDO DO I=1,9 S1=0 IF (IPERT.EQ.1) THEN !Minimum (Aphelion) perturbations from other planets (all !other planets aligned co-linearly on opposite side of Sun): DO J=1,9 IF (J.NE.I) THEN S1=S1+MASS(J)*EARTHMASS/(ABS(DISTANCE(J)+DISTANCE(I))*AU) ENDIF ENDDO ENDIF !Add Sun + impact planet (+ other planets): VGRAVA(I)=SQRT(2*G*(SUNMASS/(DISTANCE(I)*AU*(1+ECC(I)))+ & MASS(I)*EARTHMASS/(RADIUS(I)*EARTHRADIUS) & +S1)) ENDDO !Print out tables to screen and to output file "asteroid.out": WRITE(*,*)'The output to this program lists (1) planet name,' WRITE(*,*)'(2) semi-major axis (i.e., time-averaged mean' WRITE(*,*)'distance to Sun), (3) impact speed caused by' WRITE(*,*)'revolution (calculated at Perihelion and Aphelion),' WRITE(*,*)'(4) impact speed caused by gravity (calculated at' WRITE(*,*)'Perihelion and Aphelion), (5) minumum impact speed' WRITE(*,*)'(gravity minus revolution speed at Perihelion and' WRITE(*,*)'Aphelion), and (6) maximum impact speed (sum of' WRITE(*,*)'revolution and gravity-induced speeds at Perihelion' WRITE(*,*)'and Aphelion). All speeds are in km/sec. The' WRITE(*,*)'time-mean distance to the Sun is given in Astronomical' WRITE(*,*)'Units (AU; 1 AU = 1.495E11 m). The last table is a' WRITE(*,*)'summary of minimum and maximum impact speeds and' WRITE(*,*)'includes escape velocity and time-averaged mean' WRITE(*,*)'revolution speeds for each planet.' WRITE(*,*)' ' WRITE(*,*)'PERIHELION:' WRITE(*,*)' ' WRITE(*,*) &' PLANET R(AU) REV(KM/S) GRAV(KM/S) MIN(KM/S) MAX(KM/S)' WRITE(*,*) &'-------------------------------------------------------------' DO I=1,9 WRITE(*,'(A9,2X,F6.2,3X,F6.1,4X,F6.1,6X,F6.1,5X,F6.1)') & PLANETNAME(I),DISTANCE(I),VREVP(I)*0.001,VGRAVP(I)*0.001, & (VGRAVP(I)-VREVP(I))*0.001,(VGRAVP(I)+VREVP(I))*0.001 ENDDO WRITE(*,*)' ' WRITE(*,*)'APHELION:' WRITE(*,*)' ' WRITE(*,*) &' PLANET R(AU) REV(KM/S) GRAV(KM/S) MIN(KM/S) MAX(KM/S)' WRITE(*,*) &'-------------------------------------------------------------' DO I=1,9 WRITE(*,'(A9,2X,F6.2,3X,F6.1,4X,F6.1,6X,F6.1,5X,F6.1)') & PLANETNAME(I),DISTANCE(I),VREVA(I)*0.001,VGRAVA(I)*0.001, & (VGRAVA(I)-VREVA(I))*0.001,(VGRAVA(I)+VREVA(I))*0.001 ENDDO WRITE(*,*)' ' WRITE(*,*)'SUMMARY (REV is average revolution speed and VESC' WRITE(*,*)'is planetary escape velocity):' WRITE(*,*)' ' WRITE(*,*) &' PLANET R(AU) REV(KM/S) VESC(KM/S) MIN(KM/S) MAX(KM/S)' WRITE(*,*) &'-------------------------------------------------------------' DO I=1,9 WRITE(*,'(A9,2X,F6.2,3X,F6.1,4X,F6.1,6X,F6.1,5X,F6.1)') & PLANETNAME(I),DISTANCE(I),VREV(I)*0.001,VESC(I)*0.001, & MIN(VGRAVP(I)-VREVP(I),VGRAVA(I)-VREVA(I))*0.001, & MAX(VGRAVP(I)+VREVP(I),VGRAVA(I)+VREVA(I))*0.001 ENDDO IF (IPERT.EQ.1) THEN WRITE(*,*)' ' WRITE(*,*)'(NOTE: The calculated planetary impact speeds for' WRITE(*,*)'each planet in these tables include gravitational' WRITE(*,*)'perturbations from the other eight planets.)' ENDIF IF (IPERT.NE.1) THEN WRITE(*,*)' ' WRITE(*,*)'(NOTE: The calculated planetary impact speeds for' WRITE(*,*)'each planet in these tables do not include' WRITE(*,*)'gravitational perturbations from the other eight' WRITE(*,*)'planets.)' ENDIF OPEN(12,FILE='asteroid.out',FORM='FORMATTED',STATUS='UNKNOWN') WRITE(12,*)'The output to this program lists (1) planet name,' WRITE(12,*)'(2) semi-major axis (i.e., time-averaged mean' WRITE(12,*)'distance to Sun), (3) impact speed caused by' WRITE(12,*)'revolution (calculated at Perihelion and Aphelion),' WRITE(12,*)'(4) impact speed caused by gravity (calculated at' WRITE(12,*)'Perihelion and Aphelion), (5) minumum impact speed' WRITE(12,*)'(gravity minus revolution speed at Perihelion and' WRITE(12,*)'Aphelion), and (6) maximum impact speed (sum of' WRITE(12,*)'revolution and gravity-induced speeds at Perihelion' WRITE(12,*)'and Aphelion). All speeds are in km/sec. The' WRITE(12,*)'distance to the Sun is given in Astronomical Units' WRITE(12,*)'(AU; 1 AU = 1.495E11 m). The last table is a' WRITE(12,*)'summary of minimum and maximum impact speeds and' WRITE(12,*)'includes escape velocity and time-averaged mean' WRITE(12,*)'revolution speeds for each planet.' WRITE(12,*)' ' WRITE(12,*)'PERIHELION:' WRITE(12,*)' ' WRITE(12,*) &' PLANET R(AU) REV(KM/S) GRAV(KM/S) MIN(KM/S) MAX(KM/S)' WRITE(12,*) &'-------------------------------------------------------------' DO I=1,9 WRITE(12,'(A9,2X,F6.2,3X,F6.1,4X,F6.1,6X,F6.1,5X,F6.1)') & PLANETNAME(I),DISTANCE(I),VREVP(I)*0.001,VGRAVP(I)*0.001, & (VGRAVP(I)-VREVP(I))*0.001,(VGRAVP(I)+VREVP(I))*0.001 ENDDO WRITE(12,*)' ' WRITE(12,*)'APHELION:' WRITE(12,*)' ' WRITE(12,*) &' PLANET R(AU) REV(KM/S) GRAV(KM/S) MIN(KM/S) MAX(KM/S)' WRITE(12,*) &'-------------------------------------------------------------' DO I=1,9 WRITE(12,'(A9,2X,F6.2,3X,F6.1,4X,F6.1,6X,F6.1,5X,F6.1)') & PLANETNAME(I),DISTANCE(I),VREVA(I)*0.001,VGRAVA(I)*0.001, & (VGRAVA(I)-VREVA(I))*0.001,(VGRAVA(I)+VREVA(I))*0.001 ENDDO WRITE(12,*)' ' WRITE(12,*)'SUMMARY (REV is average revolution speed and VESC' WRITE(12,*)'is planetary escape velocity):' WRITE(12,*)' ' WRITE(12,*) &' PLANET R(AU) REV(KM/S) VESC(KM/S) MIN(KM/S) MAX(KM/S)' WRITE(12,*) &'-------------------------------------------------------------' DO I=1,9 WRITE(12,'(A9,2X,F6.2,3X,F6.1,4X,F6.1,6X,F6.1,5X,F6.1)') & PLANETNAME(I),DISTANCE(I),VREV(I)*0.001,VESC(I)*0.001, & MIN(VGRAVP(I)-VREVP(I),VGRAVA(I)-VREVA(I))*0.001, & MAX(VGRAVP(I)+VREVP(I),VGRAVA(I)+VREVA(I))*0.001 ENDDO IF (IPERT.EQ.1) THEN WRITE(12,*)' ' WRITE(12,*)'(NOTE: The calculated comet (asteroid, etc.)' WRITE(12,*)'impact speeds for each planet in these tables' WRITE(12,*)'include gravitational perturbations from the' WRITE(12,*)'other eight planets.)' ENDIF IF (IPERT.NE.1) THEN WRITE(12,*)' ' WRITE(12,*)'(NOTE: The calculated comet (asteroid, etc.)' WRITE(12,*)'impact speeds for each planet in these tables' WRITE(12,*)'do not include gravitational perturbations from' WRITE(12,*)'the other eight planets.)' ENDIF STOP END