How To Keep Asteroids From Crashing into Earth

Joachim Köppen Kiel/Strasbourg/Illkirch December 2001


Contents


................... under development.................


About the Applet and what it does

The Applet is more a tool to get quantitative results than to give a graphical view of the asteroid's trajectory. The basic situation here is the same as in EarthShooter and NeoSim applets. We consider how the asteroid moves in the Earth's gravitational field, neglecting the influence of Moon and Sun. However, this applet is designed to show how results - such as the perigee - depend on the initial conditions and on the details of the deflection maneuver.

In this simple two-body problem the laws of conservation of energy and angular momentum permit to completely describe the behaviour of the asteroid. These formulae are evaluated and their results are displayed as graphs. The user is able to look at all relevant quantities of the problem, while the machine takes care of the tedious maths.

Let's take first steps

When you start up the applet, you hit the Click To Start button, and you see something like the following screen:

This shows how the perigee (in Earth radii) of ten asteroids coming in with velocity of 2 km/s at impact parameters between 50 and 100 Earth radii (note: lunar orbit is at about 60 Earth radii) depends on the magnitude of a deflection maneuver.

This applet is rather complicated, because we have four independent model parameters (impact parameter, initial velocity, magnitude and direction of the velocity change appied during the maneuver) and many results (like the perigee), but only a two-dimensional screen. So we can vary smoothly only a single parameter, which we do by the choice of the horizontal axis. The other three we can only vary by giving them a number of discrete values in a range of values, and drawing curves for all the combinations of their values. The number of points as well as the lower and upper limit for the range are entered in the textfields corresponding to each variable. With the choices, we can select how we deal with these parameters. Note that for a proper calculation, all the four parameters must be given. Therefore the Show button is only enabled when your choices are correct.

The vertical axis of the plot is used for the results. But one may also plot any of the four model parameters, which has no extremely sensible meaning, but may just be helpful.

Here is an example: we consider three asteroids coming in with an initial velocity of 2 km/s and having three diffrent impact parameters of 4, 5.5, and 7 Earth radii. A maneuver takes place when the bodies are at 100 Earth radii distance (cf. the situation in the other Applet ). At that point, we change the velocity only in the direction of their flight (Delta V direction is 0), i.e. we push the asteroids from "behind" and make them faster. The amount of this velocity change makes the horizontal axis of the graph, and here we consider changes between 0 and 1 km/s. The following plot shows the effect of this Delta V on the perigee:

The blue line denotes the Earth surface, the red curves give the perigee as a function of the deflection maneuver. The top red curve, with an impact parameter of 7 Earth radii, remains always above the blue line. This means that if we did nothing at all, the asteroid would pass 0.1 Earth radii (= 670 km) above the Earth surface, which would be a very narrow miss. Giving the body a push by 0.5 km/s would raise the perigee to about 1.7 Earth radii, or some 5000 km above the surface. To avoid a collision with a body with impact parameter of 5.5. Earth radii would require a larger effort, at least about 0.4 km/s. The lowest curve refers to an impact parameter of only 4 Earth radii, and at least 1 km/s are required.

That the red curves slope upwards towards the right hand side merely reflects the fact that increasing the velocity of the object will make it pass by Earth at a greater height, and thus is helpful.

This plot shows - for the same models - that all three bodies have positive total energy after the maneuver, thus they will remain unbound, and they will leave us for good. You may verify that their initial total energy was already positive, as would be expected for a body coming from outside the region of the Earth's gravitational influence. Likewise, one can display the Delta Energy and show that the total energy increased by the maneuver.

If we consider a fixed Delta V, but look at how the angle of the maneuver affects the perigee, we note that the angle 0 - the push in flight direction - has the largest benefit! The most efficient way to defend oneself against being shot is to speed up the bullet coming towards you!

Of course, this works only if gravitational forces act between you and the bullet ... and if it would not hit you if it could fly on a perfectly straight line! If the impact parameter is smaller, i.e. if the asteroid would hit the Earth if it had infinite velocity, then a deflection maneuver has to be done with the Delta V vector pointing away from the flight direction. This means that the maneuver will not be the most efficient, and thus could be rather costly. The following plot shows what should be done if an asteroid is directly headed for the Earth centre with 2 km/s speed:

The best way is to push it perpendicularly to its flight path, with at least 0.15 km/s. The direction of the maneuver is not very critical, but the speed is.

Finally, here is an extreme case: a body at rest at 100 Earth radii away, receives a push 90 degrees off the direction towards the Earth:

The perigee increases with increasing Delta V, the orbits are ellipses, but at about 0.8 km/s it becomes constant: this is the speed for a circular orbit of 100 Earth radii. Further increase of the Delta V do not change the perigee, because first the orbit becomes elliptical again with an increasingly higher apogee, and finally the body reaches a total energy sufficient for escape from the Earth, and thus its initial position remains its perigee.

The controls:
Click to Start
Starts the applet
drag & Zoom
after clicking the button, click on the plot where you want to zoom in the upper left hand corner of the area to be zoomed, and drag the mouse down and to the right. Release of the mouse will make the zoomed view to appear
unZoom
restores the full normal view of the plot
show
displays the plot of the selected by the choices
vertical axis =
this choice selects the variable which is plotted on the vertical axis
standard range
restores the normal range of plotting the variable for the vertical axis. The text fields to the left and right show the minimum and maximum values currently used in the plot, e.g. after a zoom. You may also alter these values by clicking on the text field and entering the desired values
horizontal axis =
this choice selects the variable which is plotted on the horizontal axis
standard range
as above, but for the horizontal axis

The other three variables are dealt with in the same way:

choice
selects the variable
... points
the text field to the right of the choice gives the number of values which this variable is allowed to take when the plot is produced and for each combination of the values a curve is drawn
text fields
show the currently used minimum and maximum values for the variable. If only one point is requested for it, it will be its minimum value

The variables:
impact p. [R_E]
the impact parameter in units of Earth radii
ini.vel. [km/s]
the initial velocity in km/s
Delta V [km/s]
the amount of velocity change
Delta V direct. [deg]
the angle between the vector of velocity change and the flight direction
ini. distance [R_E]
the initial distance
beta [deg]
the angle between the initial flight direction and the direction to the Earth's centre
fin.vel. [kms]
the magnitude of the velocity after the maneuver
V direction change [deg]
the angle between the velocities before and after the maneuver
ini.Ang.Mom. []
initial angular momentum (in some convenient units)
fin.Ang.Mom. []
angular momentum after maneuver
Delta Ang.Mom. []
the change of angular momentum due to the maneuver
ini.Energy []
initial total energy of asteroid (in some convenient units)
fin.Energy []
asteroid's total energy after the maneuver
Delta Energy []
change of asteroid's total energy due to the maneuver
perigee [R_E]
closest distance to Earth's centre during fly-by (in Earth radii)
perigee alt. [km]
the altitude above Earth surface at closest approach (in km)


The Formulae behind the Screen

Let us place the Earth at the centre of a Cartesian coordinate system, and the asteroid at its initial position at (x_0, y_0). In the applet I fixed x_0 = -100 R_E (Earth radii) and so y_0 = p is the impact parameter. Thus the initial distance to the Earth's centre is r^2 = (x_0)^2 + p^2, the angle beta is given by the relation sin(beta) = p/r.
The initial total energy (per kg of the asteroid mass) is the sum of kinetic and potential energy:

E_i = (v_i)^2 / 2 - GM_E/r
with the gravitational constant G and the Earth mass M_E.
The angular momentum (also per unit mass) is
L_i = r v_i sin(beta) = p v_i.
(By the way, it is convenient to use normalized quantities: by putting GM_E = 1 and by chosing the Earth radius as the unit of length, we obtain units of time and velocity so that a satellite in circular orbit at the Earth's surface has a period of 6.28... time units and a speed of one velocity unit. I do make use of this in the applet, but convert to more practical units in the user interface.)

At the initial position we assume that the maneuver takes place. The velocity is changed by the amount Delta V pointing with an angle gamma away from the flight direction. We can compute the speed of the asteroid after the maneuver by applying the cosine rule to the triangle of the velocities and we get

(v_f)^2 = (v_i)^2 +(Delta V)^2 + 2(v_i)(Delta V)cos(gamma)
from which we can compute the final (that is: after the maneuver) energy and angular momentumL_f = r p sin(beta+gamma) as well as their changes. Also, the sine rule gives the angle by which the velocity has been changed: sin(alpha) = (Delta V)sin(gamma)/(v_f).

To compute the perigee, we note that at that moment the velocity vector will be perpendicular to the radius vector, so that the angular momentum is simply: (v_p)(r_p) which must be equal to the momentum L_f after the maneuver. We also apply the conservation of energy which states that the total energy at perigee E_p = (v_p)^2 / 2 - GM_E/(r_p) must be equal to the total energy after the maneuver which is E_f = (v_f)^2 / 2 - GM_E/r (note that the maneuver takes place at the initial distance r).

These conditions give two equations. The first one is simply

(v_p) = (L_f)/(r_p)
The second one is a quadratic equation for the perigee radius which must be discussed regarding the sign of the total (final) energy:
=== if E_f is positive, the asteroid after the maneuver is not gravitationally bound to the Earth, and it will pass by the Earth on a hyperbolic trajectory. The perigee will be
r_p = sqrt[(GM/(2E_f))^2 + (L_f)^2/(2E_f)] - GM/(2E_f)

=== if E_f is negative, the asteroid is captured in the Earth gravitational field. It will orbit Earth in an elliptical orbit, with the apogee
r_a = GM/(2(-E_f)) + sqrt[(GM/(2(-E_f)))^2 - (L_f)^2/(2(-E_f))]
and the perigee
r_p = GM/(2(-E_f)) - sqrt[(GM/(2(-E_f)))^2 - (L_f)^2/(2(-E_f))]

=== if E_f = 0 we have the parabolic trajectory, the asteroid will come as close as r_p = (L_f)^2/(2GM)

This gives all the information. We note that it is also possible to compute the direction of the major axis of the orbit, its eccentricity etc.

The applet does nothing but evaluate the above formulae, crank through the variables, and display the results graphically.


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