ANALYTICAL SOLUTIONS FOR MARTIAN NIGHTTIME OH* LAYER
M. Grygalashvyly, Leibniz-Institute of Atmospheric Physics, Kühlungsborn, Germany (gryga@iap-kborn.de),
D. S. Shaposhnikov, Moscow Institute of Physics and Technology, Moscow, Russia, A. S. Medvedev, Max
Planck Institute for Solar System Research, Göttingen, Germany, G. R. Sonnemann, Leibniz-Institute of Atmospheric Physics, Kühlungsborn, Germany, P. Hartogh, Max Planck Institute for Solar System Research, Göttingen, Germany.

Introduction:
Airglow emissions of OH* in the Earth mesopause region are used for obtaining information
about gravity wave, planetary wave and tidal parameters, chemical distributions (O and H), and temperature (trends, solar cycle effects, and annual variations). Recently, hydroxyl emissions were found in
the Martian atmosphere (Clancy et al., 2013), thus,
we can expect similar applications of this emission
for this planet.
In order to study morphology and variability of
the layer, the corresponding parameters should be
introduced. The concentration of OH* at peak and
peak altitude represent a natural choice for this purpose. For interpretation of measurements, it is desirable to establish straightforward relations between
these quantities and the ambient temperature, air
density and concentration of minor species involved
in photochemical reactions.
Analytical Approaches:
Assuming the photochemical equilibrium for excited
hydroxyl in the vicinity of OH* layer (~40-60 km) at
nighttime conditions (García-Muñoz et al., 2005)
and omitting the reaction HO2 with O as negligible
for population of OH* (Xu et al., 2012; GarcíaMuñoz et al., 2005), we start from the almost full
equation for vibrationally exited hydroxyl:

[𝑂𝑂𝑂𝑂𝑣𝑣 ] ≈

𝑓𝑓𝑣𝑣 𝑟𝑟1 [𝐻𝐻][𝑂𝑂3 ]+∑9𝑣𝑣′=𝑣𝑣+1 𝐴𝐴𝑣𝑣′ 𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝐶𝐶𝐶𝐶2 ]+
9
⎛∑ ′
𝐺𝐺 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝑁𝑁2 ]+∑9𝑣𝑣′=𝑣𝑣+1 𝐵𝐵𝑣𝑣′ 𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝑂𝑂2 ]⎞
𝑣𝑣 =𝑣𝑣+1 𝑣𝑣′ 𝑣𝑣
9
9
⎝ + ∑𝑣𝑣′ =𝑣𝑣+1 𝐷𝐷𝑣𝑣′𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝑂𝑂]+∑𝑣𝑣′=𝑣𝑣+1 𝐸𝐸𝑣𝑣′𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ � ⎠
,
𝑣𝑣−1
𝑣𝑣−1
∑ ′′ 𝐴𝐴𝑣𝑣𝑣𝑣′′ [𝐶𝐶𝑂𝑂2 ]+∑ ′′ 𝐺𝐺𝑣𝑣𝑣𝑣′′ [𝑁𝑁2 ]+
𝑣𝑣 =0
𝑣𝑣 =0
�∑𝑣𝑣−1
[𝑂𝑂 ]+∑𝑣𝑣−1
[𝑂𝑂]+∑𝑣𝑣−1
𝐷𝐷
𝐸𝐸
+�
𝐵𝐵
𝑣𝑣′′ =0 𝑣𝑣𝑣𝑣′′
𝑣𝑣′′ =0 𝑣𝑣𝑣𝑣′′ 2
𝑣𝑣′′ =0 𝑣𝑣𝑣𝑣′′
+𝑟𝑟4 (𝑣𝑣)[𝑂𝑂]

(1)

where v is vibrational number (here and after 𝑣𝑣 <
𝑣𝑣 ′ ; 𝑣𝑣 ′′ < 𝑣𝑣); fv is the nascent distributions, r – reaction rates, A, B, G, and D are the quenching coefficients for CO2, O2, N2, and O, respectively. All used
reactions and processes are collected in Table 1.
Hereafter, the square brackets denote number density
of the given chemical constituents.
Considering only main processes of production and
relaxation (reaction of O3 with H, quenching by CO2,
O2, and N2), we can simplify Eq. (1):

[𝑂𝑂𝑂𝑂𝑣𝑣 ] ≈

𝑓𝑓𝑣𝑣 𝑟𝑟1 [𝐻𝐻][𝑂𝑂3 ]+∑9𝑣𝑣′=𝑣𝑣+1 𝐴𝐴𝑣𝑣′ 𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝐶𝐶𝐶𝐶2 ]+
� 9
�
∑𝑣𝑣′=𝑣𝑣+1 𝐵𝐵𝑣𝑣′𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝑂𝑂2 ]+∑9𝑣𝑣′=𝑣𝑣+1 𝐺𝐺𝑣𝑣′𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝑁𝑁2 ]
[𝑂𝑂 ]+
∑𝑣𝑣−1
𝐵𝐵
𝐴𝐴𝑣𝑣𝑣𝑣′′ [𝐶𝐶𝑂𝑂2 ]+∑𝑣𝑣−1
′′
𝑣𝑣′′ =0 𝑣𝑣𝑣𝑣′′ 2
� 𝑣𝑣 =0
�
[𝑁𝑁
]
+ ∑𝑣𝑣−1
𝐺𝐺
𝑣𝑣′′ =0 𝑣𝑣𝑣𝑣′′ 2

. (2)

We omitted spontaneous emission and quenching by
O as much weaker processes. Indeed, for OHv=9 and
OHv=1 the total spontaneous emission coefficients
amount to E9=199.3 s-1 and E1=17.6 s-1 (Xu et al.,
2012), respectively. On the other hand, for example
at 50 km, [CO2]≥1015 cm-3 (e.g. Krasnopolsky and
Lefèvre, 2013), total collisional removal rates
A9=9.1∙10-11 cm3s-1 and A1=2.9∙10-13 cm3s-1 (e.g.
Krasnopolsky (2013), García-Muñoz et al. (2005)),
therefore the first term in the denominator of Eq. (1)
for corresponding vibrational numbers amounts to
≥9∙104 s-1 and ≥2.9∙102 s-1, respectively. [O] at 50-60
km is ~109-1011 cm-3 (e.g. Krasnopolsky and
Lefèvre, 2013; Krasnopolsky, 2010; Krasnopolsky,
2006). Caridade et al. (2013) derives total values for
and
non-reactive
reactive
(O+OHv→O2+H)
(O+OHv→OHv`<v+O) quenching rates by O at temperature 160 K as 7.7∙10-11 cm3s-1 and 6∙10-11 cm3s-1
for vibrational numbers nine and one, respectively.
Hence, the corresponding collisional removal due to
atomic oxygen ≤8-6 s-1 for all vibrational numbers,
and can be safely neglected.
Table 1. List and nomenclature of reaction rates, quench-

ing coefficients, spontaneous emission coefficients (references: 1,2,3Burkholder et al. (2020), 1,5Adler-Golden
(1997), 4,5Caridade et al. (2013), 5Makhlouf et al. (1995),
5Krasnopolsky (2013), 6 Xu et al. (2012)).
Reactions
Coefficients
−470
1 𝐻𝐻 + 𝑂𝑂3
𝑟𝑟1 = 1.4 ∙ 10−10 𝑒𝑒𝑒𝑒𝑒𝑒 �
�
𝑓𝑓𝑣𝑣 𝑟𝑟1
𝑇𝑇
�⎯� 𝑂𝑂𝑂𝑂𝑣𝑣=5,…,9 + 𝑂𝑂2
𝑓𝑓𝑣𝑣=9,…,5
= 0.47,0.34,0.15,0.03,0.01
2
𝑂𝑂 + 𝑂𝑂2 + 𝐶𝐶𝐶𝐶2
𝑟𝑟2 = 6.1 ∙ 10−34 (298⁄𝑇𝑇 )2.4
→ 𝑂𝑂3 + 𝐶𝐶𝐶𝐶2
−2060
3
𝑂𝑂 + 𝑂𝑂3 → 2𝑂𝑂2
𝑟𝑟3 = 8 ∙ 10−12 𝑒𝑒𝑒𝑒𝑒𝑒 �
�
𝑇𝑇
4 𝑂𝑂 + 𝑂𝑂𝑂𝑂𝑣𝑣=1,..,9 → 𝑂𝑂2
𝑟𝑟4 (𝑣𝑣 = 9, … ,1) = (5.42,4.8,
4.42, 4, 3.77,4.43,3.74,3,3.15)
+ 𝐻𝐻
∙ 10−11
5
𝑂𝑂𝑂𝑂𝑣𝑣
𝐴𝐴𝑣𝑣𝑣𝑣′ 𝐵𝐵𝑣𝑣𝑣𝑣′ , 𝐺𝐺𝑣𝑣𝑣𝑣′ , 𝐷𝐷𝑣𝑣𝑣𝑣′
See text
+ 𝐶𝐶𝐶𝐶2 , 𝑂𝑂2 , 𝑁𝑁2 , 𝑂𝑂
→ 𝑂𝑂𝑂𝑂𝑣𝑣′<𝑣𝑣 + 𝐶𝐶𝐶𝐶2
6
𝑂𝑂𝑂𝑂𝑣𝑣
𝐸𝐸𝑣𝑣𝑣𝑣′
→ 𝑂𝑂𝑂𝑂𝑣𝑣′<𝑣𝑣 + ℎ𝑣𝑣

Following García-Muñoz et al. (2005), we assume
O3 in the photochemical equilibrium in the vicinity
of nighttime OH* layer. Then the ozone balance

equation for nighttime conditions can be represented
as follows:
𝑟𝑟2 [𝑂𝑂][𝑂𝑂2 ][𝐶𝐶𝐶𝐶2 ] = 𝑟𝑟1 [𝑂𝑂3 ][𝐻𝐻] + 𝑟𝑟3 [𝑂𝑂][𝑂𝑂3 ].

(3)

The share of the reaction of O3 with O in total ozone
loss is small, since for typical temperatures at 50-60
km (~150 K) r3 (~8.7∙10-18 cm3s-1) is ~106 times
smaller than r1 (~6.1∙10-12 cm3s-1), but [H] smaller
than [O] no more than ~102-103 times in this region
(García-Muñoz et al., 2005; Krasnopolsky, 2006;
Krasnopolsky and Lefèvre, 2013). Thus, the second
term in the right-hand side of Eq. (3) can be omitted
and it becomes
𝑟𝑟2 [𝑂𝑂][𝑂𝑂2 ][𝐶𝐶𝐶𝐶2 ] ≈ 𝑟𝑟1 [𝑂𝑂3 ][𝐻𝐻].

(4)

Substituting Eq. (4) into the first term in the numerator of Eq. (2) we obtain

[𝑂𝑂𝑂𝑂𝑣𝑣 ] ≈

𝑓𝑓𝑣𝑣 𝑟𝑟2 [𝑂𝑂][𝑂𝑂2 ][𝐶𝐶𝐶𝐶2 ]+∑9𝑣𝑣′=𝑣𝑣+1 𝐴𝐴𝑣𝑣′ 𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝐶𝐶𝐶𝐶2 ]+

� 9
�
∑𝑣𝑣′=𝑣𝑣+1 𝐵𝐵𝑣𝑣′ 𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝑂𝑂2 ]+∑9𝑣𝑣′=𝑣𝑣+1 𝐺𝐺𝑣𝑣′𝑣𝑣 �𝑂𝑂𝑂𝑂𝑣𝑣′ �[𝑁𝑁2 ]

[𝐶𝐶𝑂𝑂2 ]+∑𝑣𝑣−1
[𝑂𝑂 ]+∑𝑣𝑣−1
[𝑁𝑁 ]
∑𝑣𝑣−1
𝐵𝐵
𝐺𝐺
𝐴𝐴
𝑣𝑣′′ =0 𝑣𝑣𝑣𝑣′′
𝑣𝑣′′ =0 𝑣𝑣𝑣𝑣′′ 2
𝑣𝑣′′ =0 𝑣𝑣𝑣𝑣′′ 2

. (5)

𝛽𝛽𝛽𝛽𝑣𝑣 𝑟𝑟2 [𝑂𝑂]𝑀𝑀 + [𝑂𝑂𝑂𝑂𝑣𝑣 ′ ] ∑9𝑣𝑣 ′=𝑣𝑣+1 𝐶𝐶𝑣𝑣′𝑣𝑣
[𝑂𝑂𝑂𝑂𝑣𝑣 ] ≈
,
(6)
∑𝑣𝑣−1
𝑣𝑣 ′′ =0 𝐶𝐶𝑣𝑣𝑣𝑣′′
9
9
9
where ∑𝑣𝑣′=𝑣𝑣+1 𝐶𝐶𝑣𝑣′𝑣𝑣 = ∑𝑣𝑣′=𝑣𝑣+1 𝐴𝐴𝑣𝑣′𝑣𝑣 + 𝛼𝛼 ∑𝑣𝑣′=𝑣𝑣+1 𝐵𝐵𝑣𝑣′𝑣𝑣 +
𝑣𝑣−1
∑𝑣𝑣−1
𝜒𝜒 ∑9𝑣𝑣 ′=𝑣𝑣+1 𝐺𝐺𝑣𝑣′𝑣𝑣
and
𝑣𝑣 ′′ =0 𝐶𝐶𝑣𝑣𝑣𝑣′′ = ∑𝑣𝑣 ′′ =0 𝐴𝐴𝑣𝑣𝑣𝑣′′ +
𝑣𝑣−1
𝑣𝑣−1
𝛼𝛼 ∑𝑣𝑣 ′′=0 𝐵𝐵𝑣𝑣𝑣𝑣′′ + 𝜒𝜒 ∑𝑣𝑣 ′′=0 𝐺𝐺𝑣𝑣𝑣𝑣′′ .

Writing reaction rate r2 explicitly and reorganizing
Eq. (6), we obtain:
[𝑂𝑂𝑂𝑂𝑣𝑣 ] ≈ 𝜀𝜀𝜀𝜀𝑣𝑣
𝜀𝜀 = 6.1 ∙ 10−24 ∙ 2982.4 𝛽𝛽

where
′

=9
𝑓𝑓𝑣𝑣 +∑𝑣𝑣𝑣𝑣′ =𝑣𝑣+1
𝛾𝛾𝑣𝑣′ 𝐶𝐶𝑣𝑣′𝑣𝑣
′′ =𝑣𝑣−1
∑𝑣𝑣𝑣𝑣′′ =0
𝐶𝐶𝑣𝑣𝑣𝑣′′

and

(7)
𝛾𝛾𝑣𝑣 =

, (𝑓𝑓9<𝑣𝑣<5 = 0). Note that the numerical

coefficient in ε, which comes from the reaction rate
r2, can be different. Nevertheless, all studies are in
consensus that 𝑟𝑟2 ~𝑇𝑇 −2.4 .
Peak altitude. Eq. (7) can be rewritten applying the
ideal gas law:
[𝑂𝑂𝑂𝑂𝑣𝑣 ] ≈ 𝜗𝜗𝑣𝑣 𝑇𝑇 −3.4 [𝑂𝑂]𝑝𝑝,
(8)
where 𝜗𝜗𝑣𝑣 = 𝜀𝜀𝛾𝛾𝑣𝑣 ⁄𝑘𝑘𝑏𝑏 , 𝑘𝑘𝑏𝑏 is Boltzmann constant, and p

is pressure.
Differentiating by pressure and equating to zero, we
derive the expression for pressure at local maximum
of the excited hydroxyl concentration:
𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚≈

1
≈ 1�
,
𝑇𝑇 3.4
𝜕𝜕
𝜕𝜕 ln 𝑇𝑇 𝜕𝜕 ln[𝑂𝑂]
�ln �
��
3.4
−
[𝑂𝑂]
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

(9)

Substituting (9) into (8), we obtain the expression
for excited hydroxyl at their maximum:
[𝑂𝑂𝑂𝑂𝑣𝑣 ]𝑚𝑚𝑚𝑚𝑚𝑚≈

𝜗𝜗𝑣𝑣 𝑇𝑇 −3.4 [𝑂𝑂]
𝜗𝜗𝑣𝑣 𝑇𝑇 −3.4 [𝑂𝑂]
≈
.
𝜕𝜕 ln 𝑇𝑇 𝜕𝜕 ln[𝑂𝑂]
𝜕𝜕
𝑇𝑇 3.4
−
3.4
�ln �
��
𝜕𝜕𝜕𝜕
[𝑂𝑂]
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

[𝑂𝑂𝑂𝑂𝑣𝑣 ] ≈ 𝜀𝜀𝜀𝜀𝑣𝑣 ����
[𝑂𝑂]𝑇𝑇� −2.4 �����
[𝑀𝑀] + 𝜀𝜀𝜀𝜀𝑣𝑣 ����
[𝑂𝑂]𝑇𝑇� −2.4 [𝑀𝑀]′ +
′ � −2.4 [𝑀𝑀]
′ � −3.4 [𝑀𝑀]
�����
����
����� +
[𝑂𝑂]
[𝑂𝑂]
𝜀𝜀𝜀𝜀𝑣𝑣
𝑇𝑇
− 2.4𝜀𝜀𝜀𝜀𝑣𝑣 𝑇𝑇 𝑇𝑇
����𝑇𝑇 ′ 𝑇𝑇� −3.4 [𝑀𝑀]′ −
𝜀𝜀𝜀𝜀𝑣𝑣 [𝑂𝑂]′ 𝑇𝑇� −2.4 [𝑀𝑀]′ − 2.4𝜀𝜀𝜀𝜀𝑣𝑣 [𝑂𝑂]
����� − 2.4𝜀𝜀𝜀𝜀𝑣𝑣 [𝑂𝑂]′ 𝑇𝑇 ′ 𝑇𝑇� −3.4 [𝑀𝑀]′ .
2.4𝜀𝜀𝜀𝜀𝑣𝑣 [𝑂𝑂]′ 𝑇𝑇 ′ 𝑇𝑇� −3.4 [𝑀𝑀]

(12)

The exited hydroxyl concentration for a given vibrational number can be written as follows:

′
′
′
��������
[𝑂𝑂𝑂𝑂𝑣𝑣 ] ≈ [𝑂𝑂𝑂𝑂
𝑣𝑣 ] + [𝑂𝑂𝑂𝑂𝑣𝑣 ] 𝑀𝑀 + [𝑂𝑂𝑂𝑂𝑣𝑣 ] 𝑂𝑂 + [𝑂𝑂𝑂𝑂𝑣𝑣 ] 𝑇𝑇 +
′′
′′
′′
[𝑂𝑂𝑂𝑂𝑣𝑣 ] 𝑂𝑂𝑂𝑂 + [𝑂𝑂𝑂𝑂𝑣𝑣 ] 𝑇𝑇𝑇𝑇 + [𝑂𝑂𝑂𝑂𝑣𝑣 ] 𝑇𝑇𝑇𝑇 + ℎ𝑖𝑖. 𝑜𝑜𝑜𝑜𝑜𝑜. 𝑡𝑡𝑡𝑡𝑡𝑡., (13)
��������
���� ∙ �����
[𝑂𝑂𝑂𝑂𝑣𝑣 ] = 𝜀𝜀𝛾𝛾𝑣𝑣 𝑇𝑇� −2.4 [𝑂𝑂]
[𝑀𝑀], [𝑂𝑂𝑂𝑂𝑣𝑣 ]′ 𝑀𝑀 =
where
−2.4
′
′
−2.4
����[𝑀𝑀] , [𝑂𝑂𝑂𝑂𝑣𝑣 ] = 𝜀𝜀𝛾𝛾𝑣𝑣 𝑇𝑇�
�����,
[𝑂𝑂]
[𝑂𝑂]′ [𝑀𝑀]
𝜀𝜀𝛾𝛾𝑣𝑣 𝑇𝑇�
𝑂𝑂
���� ∙ [𝑀𝑀]
�����, [𝑂𝑂𝑂𝑂𝑣𝑣 ]′′
[𝑂𝑂𝑂𝑂𝑣𝑣 ]′ 𝑇𝑇 = −2.4𝜀𝜀𝛾𝛾𝑣𝑣 𝑇𝑇 ′ 𝑇𝑇� −3.4 [𝑂𝑂]
=
𝑂𝑂𝑂𝑂
−2.4
′
′
′′
′
−3.4
����
�
�
[𝑂𝑂] [𝑀𝑀] , [𝑂𝑂𝑂𝑂𝑣𝑣 ] 𝑇𝑇𝑇𝑇 = −2.4𝜀𝜀𝛾𝛾𝑣𝑣 𝑇𝑇 𝑇𝑇
[𝑂𝑂][𝑀𝑀]′ ,
𝜀𝜀𝛾𝛾𝑣𝑣 𝑇𝑇
′′
′
−3.4
′
�����
�
[𝑂𝑂𝑂𝑂𝑣𝑣 ]
[𝑂𝑂] [𝑀𝑀].
= −2.4𝜀𝜀𝛾𝛾𝑣𝑣 𝑇𝑇 𝑇𝑇
𝑇𝑇𝑇𝑇

Taking into account the linear proportionality of the
molecular oxygen and molecular nitrogen number
densities to the carbon dioxide and to the concentration of air ([𝑂𝑂2 ] = 𝛼𝛼[𝐶𝐶𝐶𝐶2 ] = 𝛽𝛽𝛽𝛽, [𝑁𝑁2 ] = 𝜒𝜒[𝐶𝐶𝐶𝐶2 ]) in the
vicinity of OH* layer (e.g. Krasnopolsky, 2010;
Krasnopolsky and Lefèvre, 2013), Eq. (5) can be
rearranged as follows:

[𝑂𝑂]𝑇𝑇 −2.4 𝑀𝑀,

In Eq. (11), we apply the Taylor expansion to the
term with temperature and cross-multiply all terms:

(10)

Relative variations. Decomposing in Eq. (7) [O],
temperature, and air number density the averaged
����, 𝑇𝑇,
� 𝑀𝑀
� ) and variable parts ([𝑂𝑂]′ , 𝑇𝑇 ′ , 𝑀𝑀′) yields
([𝑂𝑂]
���� + [𝑂𝑂]′�(𝑇𝑇� + 𝑇𝑇 ′ )−2.4 �[𝑀𝑀]
����� + [𝑀𝑀]′�. (11)
[𝑂𝑂𝑂𝑂𝑣𝑣 ] ≈ 𝜀𝜀𝜀𝜀𝑣𝑣 �[𝑂𝑂]

Hence, relative variations (RV) of emissions due to
variations in temperature, [O], and concentration of
air are:
𝑅𝑅𝑅𝑅𝑇𝑇′ = 100% ∙

[𝑂𝑂𝑂𝑂𝑣𝑣 ]′𝑇𝑇
𝑇𝑇 ′
= 100% ∙ −2.4 ,
��������
𝑇𝑇�
[𝑂𝑂𝑂𝑂𝑣𝑣 ]

𝑅𝑅𝑅𝑅𝑂𝑂′ = 100% ∙

𝑅𝑅𝑅𝑅𝑀𝑀′ = 100% ∙

[𝑂𝑂𝑂𝑂𝑣𝑣 ]′𝑂𝑂
[𝑂𝑂]′
= 100% ∙
,
��������
����
[𝑂𝑂𝑂𝑂
[𝑂𝑂]
𝑣𝑣 ]

[𝑂𝑂𝑂𝑂𝑣𝑣 ]′𝑀𝑀
��������
[𝑂𝑂𝑂𝑂𝑣𝑣 ]

= 100% ∙

[𝑀𝑀]′
�����
[𝑀𝑀]

.

(14)

The relative variations of concentrations due to second momenta are
′′
𝑅𝑅𝑅𝑅𝑇𝑇𝑇𝑇
= 100% ∙

[𝑂𝑂𝑂𝑂𝑣𝑣 ]′′
𝑇𝑇𝑇𝑇
��������
[𝑂𝑂𝑂𝑂
𝑣𝑣 ]

′′
= 100% ∙
𝑅𝑅𝑅𝑅𝑇𝑇𝑇𝑇

[𝑂𝑂𝑂𝑂𝑣𝑣 ]′′
𝑇𝑇𝑇𝑇
��������
[𝑂𝑂𝑂𝑂
𝑣𝑣 ]

′′
𝑅𝑅𝑅𝑅𝑂𝑂𝑂𝑂
= 100% ∙

= 100% ∙ −2.4

[𝑂𝑂𝑂𝑂𝑣𝑣 ]′′
𝑂𝑂𝑂𝑂
��������
[𝑂𝑂𝑂𝑂
𝑣𝑣 ]

= 100% ∙

= 100% ∙

𝑇𝑇 ′ [𝑀𝑀]′
����� ,
𝑇𝑇� [𝑀𝑀]

[𝑂𝑂]′ [𝑀𝑀]′
����[𝑀𝑀]
����� ,
[𝑂𝑂]

𝑇𝑇 ′ [𝑂𝑂]′
−2.4 � ����
.
𝑇𝑇 [𝑂𝑂]

(15)

In the decompositions (14) and (15), it was implicitly
assumed that the variation of height of the OH* layer
does not exceed scale height (when we decompose
the air number density). Hence, these equations are
valid for assessment of different impacts in [OH*]
(volume emission) only for intervals where height
deviations of OH* layer from averaged height are
weak.
Calculations and Discussion:
The theoretical solutions derived above assume
for quenching and spontaneous emission processes
multi-quantum relaxation, at which transitions occur
from all vibrational levels above to all levels below.
To date, not all multi-quantum quenching coefficients for CO2 and N2 are known. Currently known
are the so-called collisional cascade quenching rates,
when for all vibrational levels transitions are assumed to one level below. Last update of these coefficients was presented by Krasnopolsky (2013) and
Makhlouf et al. (1995) for quenching by CO2 and N2,
respectively. We adopted these values, i.e. for Avv`
and Gvv` and use a diagonal matrix with values of
Krasnopolsky (2013) and Makhlouf et al. (1995) for
transitions v→v-1 and assign the non-diagonal terms
to zero.
The input profiles, were taken from the MCD

(Mars Climate Database), which is based on simulations with the Laboratoire de Météorologie Dynamique General Circulation Model (Forget et al.,
1999). The MCD contains distributions of minor
gases in the Martian atmosphere, including O3
(Lefèvre et al., 2004; 2008), that is directly involved
in OH* production, H2O (Navarro et al., 2014),
which is the principal source of odd-hydrogens, and
variations of other long-lived species (CO2, N2) involved in quenching processes (Forget et al., 1998,
2008).
Figure 2. Nighttime mean one month sliding averaged
values at peak of the OHv=2 layer calculated by Eq. (1) at
middle (40°N) latitudes: a) [OHv=2], b) height of peak, c)
[O], and d) temperature.

Figure 1. Nighttime averaged zonal mean, averaged over
70°N – 90°N, and Ls 265°-320° (in order to obtain colocation with Clancy et al. (2013) observations): (a) OHv=1,..,9,
calculated by Eq. (1) (solid lines) and calculated by Eq. (7)
(dashed lines), (b) volume emission calculated by Eq. (1)
and (7) (solid and dashed lines, respectively) for vibrational transitions 1-0 (blue), 2-1 (green), and 2-0 (red).

Figure 1a illustrates a good agreement between
[OH*] and peak altitudes calculated with the full
model (Eq. 1) and with the simplified formula (Eq.
7). The best agreement occurs near the peaks at ~4853 km. We explain differences below and above the
peaks by deviations of O3 from photochemical equilibrium in the polar night area, where O3 lifetime is
prolonged under the condition of permanent night
and downward transport of O. Although for the
modelling is better to use the full model (Eq. 1),
nevertheless Eq. (7) is useful for qualitative analysis.
For Earth OH* layer, a vertical separation OH*
by vibrational numbers (e.g. Adler-Golden, 1997) is
well known. It cannot be explained from equation
(9) (p does not depend on v). This is because we
omitted quenching by O in the loss term for OH*.
Including this term into consideration (not shown
here) results in a week vertical separation by vibrational numbers. Vertical distance between layers
with different vibrational numbers is expected to be
weaker than on Earth (as was found by Clancy et al.
(2013)). This is because for Earth`s mesopause OH*
layer, O quenching, which is responsible for separation, is comparable with O2 one, but for the Martian
atmosphere it has negligible importance compared to
CO2 quenching.
Fig. 1b shows peaks at ~48-53 km and corresponding volume emissions in a good agreement
with the Clancy et al. (2013) observations.
Equations derived above provide some predictions and can be applied for analysis, which we illustrate below.

The terrestrial OH* airglow layer demonstrates
annual and semiannual variations (e.g., Gao et al.,
2010). Similar variations can be expected from the
Martian OH* due to seasonal changes in [O], air
number density and temperature. Fig. 2 shows time
series of nighttime one-month sliding averaged values at the peak of the OHv=2 layer calculated with (1)
at middle (40°N) latitudes: a) [OHv=2], b) the height
of the peak, c) [O], and d) temperature. It is seen that
the concentration and the height of the peak at the
northern middle latitude vary seasonally with the
maxim concentrations and lowest height occurring
during the first half of the year (Ls=0°-180°). The
amplitude of the annual height variation on Mars is
more than 20 km (Fig. 2b), which by several times
exceeds that near the Earth mesopause (~5-10 km).
The figures show a clear anticorrelation between the
[OHv=2] and the height of the peak, as also follows
from (8). Since volume emission is linearly proportional to the [OH*], this points out to an anticorrelation between the emission and the height of the layer. A similar anticorrelation has also been observed
on Earth (e.g., Gao et al., 2010).
Fig. 2a and 2c demonstrate a correlation between
[O] and [OH*]. This correlation happens between
Ls~210° and 340°, where the minor maximum of
[OH*] coincides with the maximum of [O]. The correlation between the air density and the peak altitude
is even more robust, because the magnitude of seasonal variations of the air density is larger than [O].
The effects of O and air densities on the OH* layer
oppose each other. When the OH* layer is low at
summer, the air density is large, while the [O] is
small. The OH* layer moves higher at winter, the air
density decreases, but the [O] rises. In Earth`s mesosphere at high and middle latitudes, the behavior of
the OH* layer is opposite: high altitude and low
emission at summer, but lower altitude and stronger
emission at winter. This is because the main driver
for OH* layer on Earth is O, which is transported
downward in winter and upward in summer. On
Mars, the layer behavior is determined additionally
by air density variations. Seasonal changes of tem-

perature play a minor role in the annual cycle of
OH*, since it varies only by about 15 K over the year
(Fig. 2d).
In order to illustrate an assessment of the inputs
into OH* layer variability from different sources, we
calculate relative variations due to [O], temperature,
and concentration of air at 40°N. For our analyses
we use first half of the year (Ls=0°-180°) when deviations of OH* height from semi-annual averaged
height do not exceed the scale height (≈10 km). For
this example the overbar in Eq. (14) and Eq. (15)
denotes semi-annually averaged values and prime
denotes difference of actual (modeled) values from
averaged.

Figure 3. Relative variations calculated by Eq. 14 (solid
lines) and Eq. 15 (dashed lines) at 40°N, for first half of
the year.

Fig. 3 shows decomposition of OH* variation by Eq.
(14) and Eq. (15) as solid and dashed lines, respectively. The variations of temperature (red lines) play
a minor role, whereas variations due to O and air
concentrations are of the same order (variation due
to air number density is slightly larger) with action in
antiphase. The first peak of OH* at Ls~40°-50° (Fig.
2a) is determined primarily by growth of air number
density (blue line) and secondary due to temperature
decline (Fig. 2d and red line on Fig. 3). The secondary peak of [OH*] at Ls~150° (Fig. 2a) is primarily
determined by growth of [O] (green line) when declining air concentration and growing temperature
act in opposite direction. The variations due to 2nd
momenta are weaker and they do not exceed 10%.
Summary and Conclusions:
The simplified relations for OH* in the Martian
atmosphere at nighttime conditions for height of OH*
peak and concentration at peak were derived with
assumptions that: 1) total quenching by CO2, O2, and
N2 dominate over quenching by O and spontaneous
emission; 2) O3 is in photochemical equilibrium in
the vicinity of OH* layer. Under these approximations, nighttime [OH*] at peak is proportional to the
[O] and negative power of temperature. The [OH*] in
the vicinity OH* layer is directly proportional to the
pressure (i.e., inversely to the layer altitude). Hence
OH* emission, which comes almost from the vicinity
of the peak, anticorrelates with the height of the OH*
layer. OH* layer on Mars may reveal annual and

semi-annual cycles, as the result of annual (semiannual) cycles of temperature, air concentration, [O],
and their superpositions. The expressions for assessment of relative variation of OH* due to variations of temperature, [O], and concentration of air
were obtained. All these expressions can be useful
for analysis and interpretation of hydroxyl emission
observations. Eqs. (9) and (10) gives possibility to
infer altitude of OH* peak and [O] at OH* peak by
surface based airglow observations.
References:
Adler-Golden, S. (1997), J. Geophys. Res., 102,
19969–19976, doi:10.1029/97JA01622.
Burkholder, J. B., et al. (2020), JPL Publication 195, http://jpldataeval.jpl.nasa.gov.
Caridade, P. J. S. B., Horta, J.-Z. J., Varandas, A. J.
C. (2013), Atmos. Chem. Phys., 13, 1-13,
doi:10.5194/acp-13-1-2013.
Clancy, R. T., et al., (2013), Icarus, 226, 272-281,
doi:10.1016/j.icarus.2013.05.035.
Forget, F., Hourdin, F., Talagrand, O. (1998), Icarus,
131, 302-316, doi:10.1006/icar.1997.5874.
Forget, F., et al. (1999), J. Geophys. Res., 104,
24155-24176, doi:10.1029/1999JE001025.
Forget, F., Millour, E., Montabone, L., Lefevre, F.
(2008),
1447,
9106,
Bibcode:2008LPICo1447.9106F.
Gao, H., Xu, J., Wu, Q. (2010), J. Geophys. Res.
115, A06313, doi:10.1029/2009JA014641.
García-Muñoz, A., McConnell, J. C., McDade, I. C.,
Melo, S. M. L. (2005), Icarus, 176, 75–95,
doi:10.1016/j.icarus.2005.01.006.
González-Galindo, F., et al. (2013), J. Geophys.
Res., 118, 2105-2123, doi:10.1002/jgre.20150.
González-Galindo, F., et al (2015), J. Geophys. Res.,
120, 2020-2035, doi:10.1002/2015JE004925.
Krasnopolsky V. A. (2006), Icarus, 185, 153–170,
doi:10.1016/j.icarus.2006.06.003.
Krasnopolsky V. A. (2010), Icarus, 207, 638–647,
doi:10.1016/j.icarus.2009.12.036.
Krasnopolsky V. A. (2013), Planet Space Sci., 85,
78–88, doi:10.1016/j.pss.2013.05.022.
Krasnopolsky, V. A. and Lefèvre, F. (2013), Comparative Climatology of Terrestrial Planets, Univ.
of Arizona, 231–275, doi:10.2458/azu_uapress9780816530595-ch11.
Lefèvre, F., Lebonnois, S., Montmessin, F., Forget,
F. (2004), J. Geophys. Res., 109, E07004,
doi:10.1029/2004JE002268.
Lefèvre, F., et al. (2008), Nature, 454, 971-975,
doi:10.1038/nature07116.
Makhlouf, U. B., Picard, R. H., Winick, J. R. (1995),
J.
Geophys.
Res.,
100,
1128911311,
doi:1029/94JD03327.
Navarro, T., et al. (2014), J. Geophys. Res., 119,
1479-1495, doi:10.1002/2013JE004550.
Xu, J., Gao, H., Smith, A. K., Zhu, Y. (2012), J.
Geophys.
Res.,
117,
D02301,
doi:10.1029/2011JD016342.