6 Velocity and theoretical modeling

The frequency of the video frames corresponds to PAL system with 25 frames per second. Thus neighboring frames are separated by 0.04 s. We will use a relative time defined as t = n/25 in seconds, where n is the already defined frame number. If we use the end portion of the trajectory (frames 150 to 191; t=6.0 s to 7.64 s), velocity $v=13 \pm 5$ kms^-1 for t=6.8 s is resulting. This means that there is not much useful information on velocity during the end portion of the trajectory. This situation changes when we compute the average velocity from frame 2 to frame 175 (from t=0.08 s to t=7.0 s). The resulting average velocity is $v=12.8 \pm 1.1$ kms^-1 for t=3.5 s. In computing the following model of motion, ablation, and radiation, our first constraint is keeping this value of velocity. The second constraint is keeping the observed heights and locations derived geometrically in the preceding section. And the third constraint is copying the observed brightness of the bolide as close as possible.

We performed only a schematic photometry using sizes of the image in the video frames transformed to absolute stellar magnitudes (distance 100 km). We assumed that the part of the steady brightness before the maximum at frame 90 was of about apparent magnitude -15. We also assumed that the limit of the camera recording was at about magnitude -3. The resulting observed light curve is compared in Fig. 3 with the modeled light curve.

  

Figure 3: Light curve of the bolide. Absolute magnitudes are plotted against heights. Thick line corresponds to the video camera observation; thin line corresponds to the model derived magnitudes

Our model is based on gross-fragmentation model (Ceplecha et al. 1993) for computing the motion and ablation of the main body, and on classical luminous equation, $I=-(\tau v^2 / 2) {\rm d}m/{\rm d}t$,modified for time elapsed from fragmentation point to the instant when the fragments stopped to contribute to luminosity, I, of the main body (they are either visible as separate objects or already not radiating enough to be detected). The luminous efficiency, $\tau$, we used takes care of the new calibration derived from the Lost City bolide and meteorite fall (Ceplecha 1996). Luminous efficiency was taken as a function of mass, m, in our model: for masses below 10 g luminous efficiency is given by the experimental values derived by Ayers et al. (1970), and for masses over 100 kg luminous efficiency is given by the new Lost City calibration. For masses between these two values, luminous efficiency was computed by linear interpolation in logarithms (corresponding to linear interpolation in stellar magnitudes). The velocity dependence of luminous efficiency was kept the same as given by Ceplecha & McCrosky (1976, Table 1).

  

Table 4: Trajectory of the bolide: geographical coordinates, heights and velocities for the main body at individual video frames; standard deviations of velocities from frame 1 to frame 191 are about 10% of their values

In modeling our case we assumed ablation coefficient $\sigma = 0.014$ s²km^-2, the average observed value for type I bolides (stone). Any attempt to use larger values was unsuccessful because it was not possible to bring the body so deep into the atmosphere with such a grazing trajectory and to keep it moving with enough mass for such a long time interval. We then assumed different initial masses, initial velocities, location of fragmentation points, and amounts of fragmented mass so that we kept the observed heights and distances along the trajectory, and luminosity as observed. The resulting values of velocities are given in Table 4, where time, t, is coordinated to individual video frames, and the geographical coordinates, $\lambda$, $\varphi$,and heights, h, are those derived in the previous Sect. by combination of the video record with the visual observations from other locations. Observed heights and observed geographical coordinates of the bolide at instants of individual frames well correspond to model computed velocities, v, and to the average velocity derived directly from observed distances ($12.8 \pm 1.1$ kms^-1). The initial velocity (outside the atmosphere), $v_\infty$, corresponding to values in Table 4 resulted as
$v_\infty = 15.1 \pm 1.4$ kms^-1.
The initial mass (before ablation and fragmentation started), $m_\infty$,and mass, $m_{\rm B}$, at the start of the video recording (frame 2) resulted as
$m_\infty = 9100$ kg and $m_{\rm B} = 6500$ kg.

  

Figure 4: Mass of the main body as resulted from the model is plotted against time

The fragmentation history can be seen in Fig. 4 where logarithm of mass of the main body is plotted as a function of time. The sudden changes in mass of the main body are just the resulting fragmentation points. They also correspond to releasing of individual larger fragments seen behind the main body on the video frames. Maximum dynamic pressure the body encountered during the modeled entry was equal to 1 MPa and was achieved just at the point of maximum brightness, where the first larger gross-fragmentation happened. We were not able to model an early fragmentation before the maximum light and above 37 km height, but this should form only a small correction to the initial mass, which might be larger by about 10% than the above given value.

The point denoted "end" in Table 4 was computed as extrapolation after frame 191 down to 3 kms^-1 (velocity at which ablation stops) keeping the same value for the ablation coefficient. This computation also yielded the value of the terminal mass as 0.94 kg. Thus one would expect meteorite falls of kilogram masses. The upper limit of a possible meteorite should not exceed 10 kg.

We also applied a single body model to all the resulting values of Table 4 in order to check our model by such a "first approximation". This way we received also an "average" apparent ablation coefficient $\sigma = 0.11$ s²km^-2 documenting the high degree of fragmentation of the meteoroid. This value is quite similar to the value of Peekskill's bolide and meteorite fall (Brown et al. 1994; Ceplecha et al. 1996) derived also from a single body model. The SP960614 bolide resembles the Peekskill bolide also by its Earth grazing trajectory.