Pre-Phase A Report - Lisa - Nasa

98 Chapter 4 Measurement Sensitivity
It is useful to approximate 32
13 (I1 +I2) byH(zc), where H(zc) is defined as H(zc) =2zc for
zc < 1/2, and H(zc) =1forzc > 1/2. H(zc) isamaximumof41%toohighatzc =0.5,
but is a very good approximation for both low and high values of zc. Thus (S/N) 2 can
be approximated by the following expressions:
(S/N) 2 = 13
16 πωc
2 GM
RV ac
(S/N) 2 = 13 (GM)2
π
32 VR3 (ac) 2
for RV/(2 ωc) .
(4.35)
Even if the event time were known, the signal would not be detectable unless S/N > 3.
The differential rate r(M) of small-body events with S/N > 3is
r(M) =πR 2 (S/N=3)F (M) , (4.36)
where F (M) is the differential flux of minor bodies and dust grains. From Eqs. (4.35),
with V =2×10 4 m/s, we get
πR 2 (S/N=3) = 5.2×10−3 M 2 for R5.3×10 5 m .
(4.37)
The flux of minor bodies and dust grains with masses less than about 0.1 kg is given by
Grun et al. [104]. The results from their Table 1 can be fit for the higher part of the mass
range by
I(M) =2.1×10 −19 M −1.34 m −2 s −1 , (4.38)
where I(M) is the integral flux of all bodies with masses greater than M. This expression
gives a good approximation to their results down to about 10−9 kg, but is several orders
of magnitude too high at lower masses.
For higher masses, estimates of the integral flux versus mass from Shoemaker [105] can
be used. They are based on counts of impact craters versus size on the Moon, and careful
analysis of the relation between crater size and the energy of the impacting body. The
results appear to be consistent within the uncertainties with those from other sources of
information.
Figure 1 of reference [105] gives an estimated curve represented by large black dots for
the cumulative frequency per year of impacts on the Earth versus the equivalent energy
of the impacts in megatons of TNT. Since 1 megaton of TNT is equivalent to 4.2×1015 J,
for a typical impact velocity of 2×104 m/s, 1 megaton corresponds to an impact by a
mass of 2.1×107 kg. Thus the results of Figure 1 of reference [105] can be converted
to the integral flux of bodies with masses between roughly 2×104 and 2×1012 kg in the
neighborhood of the Earth’s orbit.
Shoemaker’s results can be approximated by the following two power law expressions for
the integral flux:
I(M) = 4.2×10 −18 M −0.88 m −2 s −1
I(M) = 1.7×10 −21 M −0.55 m −2 s −1
for 2×10 4 kg