Green Ghost Model
With the assumption that the green ghost emissions are by the auroral green line at 557.7 nm, we can formulate a simple model. The 557.7 nm emission is a forbidden atomic oxygen emission with a radiative life-time of 0.7 s, and it is easy to excite (it only requires 4.2 eV). The long radiative life time relative to the camera exposure time, 33 ms, means that if an energy source is turned on, the emission will be observed to rise gradually towards a steady state rate, and conversely, if the source is turned off, the emission will decay with the same time constant.
Sprites occur in the atmosphere and quenching through collisions with atmospheric constituents adds an additional process for deactivating excited atoms. The net effect is that the time constant associated with the emission is lower than 0.7 s. Vallance Jones (1974) has an extensive treatment of the processes affecting the green emission, including quenching, and all reaction rate constants are given there. The dominant quenching process is collision with O2 or O. In some references N2 and NO are mentioned as possibilities, but Vallance Jones discounts that. The associated quenching coefficients vary substantially. In particular the quenching rate for O is likely larger than the value quoted by Vallance Jones (Slanger and Black, 1981), but this will not qualitatively change the model results.
A sprite is caused by an electric field set up by a lightning strike, and it decays on varying time scales as evidenced by the duration of the sprite optical emissions; some sprites last less than 10 ms and other have optical emissions lasting up to about 100 ms. With a decaying background electric field capable of exciting atomic oxygen and the 0.7 s radiative life time of atomic oxygen we have the basic elements for a model.
We use the formulation, parameter designations, and constants given by Vallance Jones (1974) in section 4.2 to calculate the 557.7 nm emission rate.
Let n be the number of exited oxygen atoms, then the 557.7 emission rate I5577 is:
I5577 = A32 * n (1)
where A32 is the radiative probability.
The rate of change in n is given by:
dn/dt = s(t) – ( A + d3 ) * n (2)
Here s(t) is the rate at which oxygen atoms are excited and (A + d3) are the losses; A is the radiative losses and d3 the quenching losses.
The energy state leading to the 557.7 emission is the O(1S) state which decays into either the O(1P) state or the ground-state O(3P) with the probabilities A32 and A31 respectively. Thus:
A = A32 + A31 (3)
The quenching loss rate, d3, is for quenching with O and O2:
d3 = k3 * [O2] + k4 * [O] (4)
where k3 and k4 are the rate constants and [O2] and [O] the altitude dependent atmospheric densities.
The values of the parameters A32, A31, k3, and k4 are given by Vallance Jones (1974):
A32 = 1.28 /s
A31 = 0.078 /s
k3 = 3E-13 cm3/s
k4 = 7.5E-12 cm3/s
The atmospheric densities [O2] and [O] are from the MSIS-E-90 Atmosphere Model provided by NASA/GSFC here (https://ccmc.gsfc.nasa.gov/modelweb/models/msis_vitmo.php). At an altitude of 95 km MSIS gives:
[O2] = 4.553E+12 /cm3
[O] = 2.611E+11 /cm3
The excitation rate, s(t), of the atomic oxygen is associated with the sprite, and we will assume the excitation is through local free electrons energized by the sprite background electric field. We will further assume that the background E-filed is set up at sprite onset and then decays exponentially with a time constant of tE0, and that the atomic oxygen excitation rate is proportional to the background electric field. The time dependent excitation rate, s(t), can then be formulated as:
s(t) = C exp(t/tE0) (5)
where C is the initial excitation value.
Our green ghost model is now defined quantitatively. The values C and tE0 are ‘free’ parameters used to fit model prediction with observations.