vmperception.l {vismeteor}

Laplace-Transformed Perception Probabilities of Visual Meteor Magnitudes

Description

Provides the Laplace-transformed perception probability of visual meteor magnitudes and its first derivative.

Usage

vmperception.l(s, deriv.degree = 0L)

Arguments

s

numerical; Real (non-complex) parameter for the Laplace transformation.

deriv.degree

integer; degree of derivative of the transformation. Currently, valid values of deriv.degree are 0, 1 and 2.

Details

The Laplace-transformed perception probabilities F(s), given as

F(s) = \mathcal{L} \left\{p\right\}(s) = \int_{-0.5}^{\infty} \, f(m) \, \mathrm e^{-s \, m} \,\mathrm{d}m \,,

are approximately

P(s) = \begin{cases} s^{-1} \, \exp\left(-4.11 \, s + 1.32 \, s^2 - 0.15 \, s^3\right)\ & \text{ if } s >= 0.0,\\ \text{undefined} \ & \text{ otherwise.} \end{cases}

Here, m is the difference between the limiting magnitude and the meteor magnitude, and f(m) denotes the perception probabilities as a function of m. The \mathcal{L} recalls here the one-sided Laplace transform.

The Laplace transform is notably effective for determining the mean and variance of observed meteor magnitudes, which are measured relative to the limiting magnitude. This is just one example of its application. This approach is valid only when the actual magnitude distribution adheres to p(m) \sim r^{-m}, where s = \log(r). In this scenario, the mean of the observable meteor magnitudes is given by -\mathcal{L}'/\mathcal{L}, and their variance is calculated as \mathcal{L}''/\mathcal{L} - (\mathcal{L}'/\mathcal{L})^2.

Value

returns the Laplace-transformed perception probabilities. If deriv.degree is specified, it will return the deriv.degree-th order derivative of these Laplace-transformed values.

See Also

vmperception vmgeom

Examples

r <- 2.0
s <- log(r)
F0 <- vmperception.l(s)
F1 <- vmperception.l(s, deriv.degree=1L)
# magnitude mean
-F1/F0
F2 <- vmperception.l(s, deriv.degree=2L)
# magnitude variance
F2/F0 - (F1/F0)^2
# plot the Laplace-transformed perception probabilities
old_par <- par(mfrow = c(1,1))
plot(
    vmperception.l,
    0.2, 1.1,
    main = paste(
        'Laplace-transformed perception',
        'probability of visual meteor magnitudes'
    ),
    col = "blue",
    log = 'y',
    xlab = 's',
    ylab = 'L'
)
par(old_par)