vismeteor (version 2.0.1)

mideal: Ideal Distribution of Meteor Magnitudes

Description

Density, distribution function, quantile function and random generation for the ideal distribution of meteor magnitudes.

Usage

dmideal(m, psi = 0, log = FALSE)
pmideal(m, psi = 0, lower.tail = TRUE, log = FALSE)
qmideal(p, psi = 0, lower.tail = TRUE)
rmideal(n, psi = 0)

Value

dmideal gives the density, pmideal gives the distribution function, qmideal gives the quantile function and rmideal generates random deviates.

The length of the result is determined by n for rmideal, and is the maximum of the lengths of the numerical vector arguments for the other functions.

qmideal can return NaN value with a warning.

Arguments

m

numeric; meteor magnitude.

psi

numeric; the location parameter of a probability distribution. It is the only parameter of the distribution.

log

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default) probabilities are \(P[M \le m]\), otherwise, \(P[M > m]\).

p

numeric; probability.

n

numeric; count of meteor magnitudes.

Details

The density of the ideal distribution of meteor magnitudes is $$ {\displaystyle \frac{\mathrm{d}p}{\mathrm{d}m} = \frac{3}{2} \, \log(r) \sqrt{\frac{r^{3 \, \psi + 2 \, m}}{(r^\psi + r^m)^5}}} $$ where \(m\) is the meteor magnitude, \(r = 10^{0.4} \approx 2.51189 \dots\) is a constant and \(\psi\) is the only parameter of this magnitude distribution.

References

Richter, J. (2018) About the mass and magnitude distributions of meteor showers. WGN, Journal of the International Meteor Organization, vol. 46, no. 1, p. 34-38

Examples

old_par <- par(mfrow = c(2,2))
psi <- 5.0
plot(
    function(m) dmideal(m, psi, log = FALSE),
    -5, 10,
    main = paste0('Density of the Ideal Meteor Magnitude\nDistribution (psi = ', psi, ')'),
    col = "blue",
    xlab = 'm',
    ylab = 'dp/dm'
)
abline(v=psi, col="red")

plot(
    function(m) dmideal(m, psi, log = TRUE),
    -5, 10,
    main = paste0('Density of the Ideal Meteor Magnitude\nDistribution (psi = ', psi, ')'),
    col = "blue",
    xlab = 'm',
    ylab = 'log( dp/dm )'
)
abline(v=psi, col="red")

plot(
    function(m) pmideal(m, psi),
    -5, 10,
    main = paste0('Probability of the Ideal Meteor Magnitude\nDistribution (psi = ', psi, ')'),
    col = "blue",
    xlab = 'm',
    ylab = 'p'
)
abline(v=psi, col="red")

plot(
    function(p) qmideal(p, psi),
    0.01, 0.99,
    main = paste('Quantile of the Ideal Meteor Magnitude\nDistribution (psi = ', psi, ')'),
    col = "blue",
    xlab = 'p',
    ylab = 'm'
)
abline(h=psi, col="red")

# generate random meteor magnitudes
m <- rmideal(1000, psi)

# log likelihood function
llr <- function(psi) {
    -sum(dmideal(m, psi, log=TRUE))
}

# maximum likelihood estimation (MLE) of psi
est <- optim(2, llr, method='Brent', lower=0, upper=8, hessian=TRUE)

# estimations
est$par # mean of psi
sqrt(1/est$hessian[1][1]) # standard deviation of psi

par(old_par)