#  Explanation of the Special Theory of Relativity by Analytical Geometry and
Reformulation of the Inverse-Square-Law

Giuseppina Modestino<EMAIL_ADDRESS>INFN, Laboratori Nazionali di
Frascati, via Enrico Fermi 40, I-00044, Frascati (Roma) Italy

###### Abstract

The space-time length $R$ between a moving source and the observation point is
calculated in order to substitute with it the spatial distance $D$, normally
used in the Newton’s law of gravitation, as well as in any
$inverse$-$square$-$law$. Fundamentally, three space-time amounts describe
dynamics. The relationship between position and field intensity is analytic,
estimable in euclidean space, and considering a linear reference system for
the time parameter. The formulation shows compatibility with fundamental rules
of classical mechanics, highlighting also hitherto unknown properties, as a
perfect analogy between morphological and physical parameters, such as the
complete correspondence between the eccentricity and the momentum in the
orbital motion. Moreover, the procedure naturally contains relativistic
formulation without introducing any special hypothesis on light speed
isotropy, asking so the question about the actual need to introduce the
concept of space-time curvature for the correct interpretation of physics
phenomena.

##  I Introduction

As physical theory, special relativity (SR) ein1  has been experimentally well
tested, and until now inconsistencies don’t result. Principally generated for
explain the results Michelson-Morley experiment mimo , SR is built on two
postulates, one regarding the invariance of the physical laws with respect to
the inertial systems, the other one regarding the constant speed of light
observed from any point of view, even if source and observer are relatively
moving. Consequently, for fitting with the real experience, the concept of
space-time curvature is introduced. The scientific literature is full of
experimental evidences supporting the relativistic hypotheses (what  and
references within). Nevertheless, SR remains a model-dependent environment,
since observations do not have an independent status from the theoretical
content in which they appear. At least, observation results are used for
replace the Einstein’s postulates, anyway with the limitation of inductive
method, for instance as done by Robertson robe . The present study concerns
fundamentally typical relativistic problems as the signal evaluation from a
moving source but regardless Einstein’s postulates neither other conventional
procedures. Synthesized by $R$, the new distance concept will be compared to
the classical one represented by $D_{t}$, and it will be applied to classical
mechanics laws, thus proving its effectiveness.

##  II moving signal observation

###  II.1 Reference system $S(x,y)$

Let’s consider a constant power source that emits a signal propagating at
radial velocity $c$ luce . It is a point source, in such a way no interference
phenomenon will occur. The source moves with constant velocity $v$ in a
reference system $S(x,y)$ (choosing a two-dimensional space with z = 0 ) in
which an observer is placed at $[0,y_{0}]$. The motion occurs in a direction
parallel to the x-axis, being $[x_{t},0]$ the coordinates of the source, and
${\bf v_{0}}\equiv[v_{0},0]$. The distance between source and observer is

$D_{t}=\sqrt{x_{t}^{2}+y_{0}^{2}}.$ (1)

The framework is illustrated in fig.1

###  II.2 Reference time line $T$

The position of the source is determined by the relationship

$x_{t}=x_{A}+v_{0}t_{A}~{}~{}~{}~{}~{}~{}~{}~{}{\text{i}f}~{}~{}~{}~{}t_{A}~{}\in
T_{A}$ (2)

where $x_{A}$ is the source position at time $t=0$, and $T_{A}$ is the
absolute reference time line, with origin at $0$. Let’s perform a translation
on the temporal reference system

$T_{A}\rightarrow T,$ (3)

that is adding the constant length to the origin (fig.2)

$t=t_{A}+x_{A}/v_{0},$ (4)

such that the temporal law of eq.2 becomes

$x_{t}=v_{0}t~{}~{}~{}~{}~{}~{}~{}~{}\text{being}~{}~{}~{}~{}t~{}\in T.$ (5)

In spite of the triviality of the last step, it is important underline that
the choice of time reference system is crucial since it must remain unchanged
in all developments of the system dynamics.

###  II.3 Intensity field and R parameter

Due to the emission, an intensity field $E_{t}$ is recorded at $[0,y_{0}]$.
The purpose of the following calculation is to find the relationship between
$x_{t}$ and $E_{t}$ within the defined space, and respecting the physical
rates $v_{0}$ and $c$. Assuming worth the $inverse~{}square~{}law$, the
intensity is

$E_{t}\propto 1/R^{2},$ (6)

where

$R\equiv c\Delta t,~{}~{}~{}~{}{\text{ and}}~{}~{}~{}~{}\Delta t\equiv
t-t_{0}$ (7)

is the time interval during which the signal is born, at $t_{0}$, and grows at
the speed $c$ until the time $t$, when it reaches the observer. Having well
defined the space-time relationship in the eq.5, now to establish a correct
prediction of $E_{t}$ is possible as a function of time, or equivalently of
the spatial coordinates of the chosen reference system. When both observer and
source are at rest, that is also $v_{0}=0$, $R$ simply corresponds to space
distance $D_{t}$, then the Coulomb’s law applies, i.e. $E_{t}\sim
1/D_{t}=const.$ Treating a dynamic case, the correspondence is not satisfied.
In that case, it is useful to consider a previous time instant

$t_{p}\equiv t-D_{t}/c~{}~{}~{}{\text{w}ith}~{}t_{p}\in T,$ (8)

and relative source position

$x_{p}\equiv v_{0}t_{p},$ (9)

such that

$x_{t}=x_{p}+\beta_{0}D_{t}=x_{p}+\beta_{0}\sqrt{x_{t}^{2}+y_{0}^{2}}$ (10)

and

$t-t_{p}=D_{t}/c=\frac{\sqrt{v_{0}^{2}t^{2}+y_{0}^{2}}}{c}$ (11)

with $\beta_{0}\equiv v_{0}/c$.

Figure 1: Coordinates of a source emitting a radial signal at light speed c,
and the observer point, respectively $[x_{t},0]$ and $[0,y_{0}]$. The source
is moving at constant velocity $v_{0}$ . The coordinate $x_{A}$ represents the
source position at time $t=0$ in the absolute reference line $T_{A}$, or
equivalently, at $t=x_{A}/v_{0}$ in the translate reference line $T$ (see the
next figure).  Figure 2: Time line on which a reference translation from
$T_{A}$ to $T$ has been performed in such a way the eq.5 holds.

The previous quadratic equation can be solved obtaining two terms

$t_{0}=\frac{t_{p}-\sqrt{\beta_{0}^{2}t_{p}^{2}+y_{0}^{2}(1-\beta_{0}^{2})/c^{2}}}{1-\beta_{0}^{2}}$
(12)

and

$t=\frac{t_{p}+\sqrt{\beta_{0}^{2}t_{p}^{2}+y_{0}^{2}(1-\beta_{0}^{2})/c^{2}}}{1-\beta_{0}^{2}}.$
(13)

Respectively, the two solutions $t_{0}$ and $t$ represent the time birth of
the signal and time observation at $[0,y_{0}]$ point. So, their difference
corresponds to the interval $\Delta t$ as defined by eq.7. The source
corresponding positions $x_{0}$ and $x_{t}$ are illustrated in fig.3. They
represent also the analytic solutions of the quadratic eq.10. So they can be
written in the following way

$x_{0,t}=\frac{x_{p}\pm\beta_{0}\sqrt{x_{p}^{2}+y_{0}^{2}(1-\beta_{0}^{2})}}{1-\beta_{0}^{2}}$
(14)

Placing $D_{0}\equiv\sqrt{x_{0}^{2}+y_{0}^{2}}$ and performing some algebra,
the following formulas are obtained

$R=D_{t}+D_{0}=$ (15) $=2\gamma_{0}^{2}(D_{t}-\beta_{0}x_{t})=$ (16)
$=2\gamma_{0}\sqrt{x_{p}^{2}\gamma_{0}^{2}+y_{0}^{2}}$ (17)

where $\gamma_{0}\equiv 1/\sqrt{1-\beta_{0}^{2}}$.
The terms $x_{t}$ and $x_{0}$ are also related by

$x_{t}-x_{0}=v_{0}(t-t_{0})=\beta_{0}R.$ (18)

The space quantities defined in the present section are illustrated in the
fig.3, while the three time parameters are shown in the fig.4.

Figure 3:  The figure shows all the spatial parameters defined in section
II.3. The point $O_{R}$ is a the virtual origin of the path $R$ covered by the
signal, at $c$ speed.  Figure 4:  The figure shows the three time parameters
$t_{0}$, $t_{p}$ and $t$ as defined in section II.3, coherently to the example
of fig.3.

In the expression of R, in particular in 17, it is easy to recognize the same
result as Feynman fey , obtained referring it to the $retarded~{}potential$ of
Liénard-Wiechert. In the same reference, how formalism is directly derived
from Maxwell’s laws is shown, and the author also reaffirms that the same
procedure was used to get to the Lorentz transformations (LTs). Although
expressing the same formalism, a direct comparison between the LTs and the
relationships from the present calculation is difficult. The problem is that
Lorentz considers two independent instants ($t$ and $t^{\prime}$), while in
this treatments there are different time values but they refer to same
physical state recorded at instant $t$. They are $t_{0}$, the time origin of
$R$, $t$ the signal recording time, and $t_{p}$, an intermediate instant
representing the time origin of $D_{t}$. Although $t_{0}$ and $t$ are linked
by a linear relationship as defined by 7, this is not true among $t_{p}$ and
the other two values, as it shown by the eqs.12 and 13. Accordingly, confusing
the time values or their mutual roles would alter the geometric rates and even
predictions about the system dynamics. To avoid such ambiguities in the laws
of physics, then we must consider the amount $R$ instead of the simple
distance $D_{t}$, since the first one is generated by a time linear
relationship 7, while $D_{t}$ is defined by the quadratic eq.11.

Figure 5:  Emitting a signal at $c$ speed, a source travels at velocity $\bf
v$. At time $t$ the distance between observer$\bf 1$ and source$\bf 2$, is
$D_{t}$. The lengths $D_{t}$ and $X_{t}$ are invariant with respect to any
reference system, as well as the corresponding value of $R$ (eq.22), thus
respecting the Galilean invariance.

##  III invariance and generalization of the reference system

Let’s consider a reference system $S(x,y)$ as in the fig.5, where the source
has coordinates $(x_{1},y_{1})$ and the observer $(x_{2},y_{2})$. In a general
case, before interaction or simple observation, both the source as the
detector can move at $\bf v_{1}$ and $\bf v_{2}$, each one following its own
physical law. The difference between the velocities

${\bf v}\equiv{\bf v_{1}}-{\bf v_{2}}$ (19)

then represents the relative velocity between the objects belonging to the
non-perturbed system. Due to movement of both points $\bf 1$ and $\bf 2$, the
distance $D_{t}$ will vary following a double time law. To express that, it is
useful to define the couple $X_{t},Y_{t}$ both depending on time, such as

$X_{t}\equiv D_{t}\cos\theta~{}~{}~{}{\text{and}}~{}~{}~{}Y_{t}\equiv
D_{t}\sin\theta$ (20)

where $\theta$ is the angle between ${\bf v}$ and $D_{t}$, so

$D_{t}=\sqrt{X_{t}^{2}+Y_{t}^{2}}$ (21)

Despite the quantities $X_{t}$ and $Y_{t}$ obey to an analytical geometry law,
the locus $(X_{t},Y_{t})$ can not be assumed as cartesian coordinate system,
given that its origin would vary instant by instant. Rather, they identify a
virtual space $\Phi(X,Y)$ where the following relationships are valid

$R(t_{0},t)\equiv c(t-t_{0})=2\gamma^{2}(D_{t}-\beta X_{t})$ (22)

with

$\beta\equiv\frac{|{\bf v}|}{c}~{}~{}~{}{\text{and}}~{}~{}~{}\gamma\equiv
1/\sqrt{1-\beta^{2}}.$ (23)

As such, the expression for $R$ can be rewritten in any reference frame while
maintaining the relationship between the basic parameters $D_{t}$, $X_{t}$ and
$Y_{t}$, and thus respecting the Galilean invariance.

##  IV  examples of space-time laws

Determining $D_{t}$, the degrees of freedom are many as are the dimensions of
space, so three in general. Each one follows its temporal law. In this
representation we have just two, $X_{t}$ and $Y_{t}$, having chosen a two-
dimensional space (with z = 0). For the complete resolution of the eq.22 and
so to define $R$, it is also necessary to know $\bf v$ which generally depends
on the initial conditions. Following, some typical field configurations.

###  IV.1 Electromagnetic signal observation

In the section II, observing the signal from a moving source, a linear law had
chosen for $X_{t}$, and a constant term for $Y_{t}$. Indeed, being $v_{0}$
constant in magnitude and direction, and remembering that by definition
$X_{t}$ is parallel to $\bf v$, the choice of the reference system

$S(x,y)\equiv\Phi(X,Y)$ (24)

is not only possible but even natural. Thus

$X_{t}=x_{t}=v_{0}t~{}~{}~{}{\text{and}}~{}~{}~{}Y_{t}=y_{0}=constant.$ (25)

Assuming this configuration and the eq.22 (or equivalently eq.7), we outline
the intensity field, with a few numerical example. In fig.6, considering both
$E\sim 1/R^{2}$ and $E\sim 1/D_{t}^{2}$, the time behaviors are shown for
three values of $\gamma_{0}$, $1.01,~{}1.51,~{}400$. The example refers to the
case with $v_{0}=\approx 0.6c$, considering $c$ as light speeddata , and being
the detector fixed to $y=0.3m$. It is evident the difference between the two
predictions, as to $\gamma_{0}$ grows. In our opinion, $E\sim 1/R^{2}$ is
right, thus confirming the significance of $t$ as present time and the
significance of $R$ as the real signal length. For $\gamma_{0}=400$, the
behavior of $R$ is calculated considering three values for the detector
distances $[0,y_{0}]$. The fig.7 shows how this geometry parameter is
physically critical, since some difference not only can generate distant
signals many orders of magnitude between them, but also morphologically very
different.

Figure 6:  The time patterns of field intensity $E\sim 1/R^{2}$ (solid line)
are shown for three values of $\gamma_{0}$, from top respectively
$1.01,~{}1.51,~{}400$. The dotted line is the prevision based on $E\sim
1/D_{t}^{2}$. The difference between them is evident to growing of the
$\gamma$ parameter. The detector position is fixed at $[0,30cm]$.  Figure 7:
The figure refers to the case $\gamma=400$, but with three different lengths
for $[0,y]$, the detector ordinates. The levels of $R$, $a$ $b$ and $c$,
correspond to $3cm$, $30cm$ as the third part of the previous example, and
$3m$. Non-linear effects are impressive especially changing the $x_{t}$ sign.

###  IV.2 Stationary orbit

The present study can be applied to electromagnetic case as well as to a
gravitational source, since both cases follow the inverse-square-law, as
described by the relation 6. In particular, we presume a body of mass $m$,
running around a gravitational source with mass $M>>m$. According to the law
6, there is a potential energy difference $\Delta U\sim 1/R$ between the two
masses. Assuming that the two objects are moving keeping constant the
potential energy difference, we obtain

$\Delta U\sim 1/R=constant\Rightarrow R=constant.$ (26)

Initially, before interacting, $M$ and $m$ moved at $\bf v_{M}$ and $\bf
v_{m}$ whose difference was $\bf v$. Keeping constant $R$ and $\beta=|{\bf
v}|/c$, it is possible verify that in the reference system $\Phi(X,Y)$, the
locus ($X_{t},Y_{t}$) which satisfies the eq.22, is an ellipse with the
geometric parameters $a$, $b$ and $c$, respectively major and minor semiaxis,
and half distance between the two foci, equal to

$a=R/2~{}~{}~{}~{}b=R/2\gamma~{}~{}~{}~{}c=\beta R/2,$ (27)

as shown in figs.8 and 9.

Figure 8:  Equipotential orbit described by the object with mass $m$ around
$M$. It is the set of $(X_{t},Y_{t})$ points solving the eq.22 with constant
$R$, and it coincides with an elliptical orbit with specific parameters as
indicated by the statements 27-29.

One of two foci is centered into the gravity center, or equivalently in the
origin of coordinate system $\Phi(X,Y)$,

$F_{1}\equiv(0,0),$ (28)

further, the ellipse eccentricity $\epsilon$ results

$\epsilon=\beta.$ (29)

Figure 9:  The same phenomenology of fig.8 is shown adding the locus
$(X_{0},Y_{0})$, dotted elliptical line on the left, solving
$X_{0}=X_{t}-\beta R$, and $R=2\gamma^{2}(D_{0}+\beta X_{0})$.

These statements confirm the Kepler’s first law, and also reveal a strong
analogy between geometrical and physical properties in such dynamics.

###  IV.3 Local Solar System

To test the effectiveness of eq.22 and 27-29, we apply them to a real case,
for instance, to the local solar system constituted by the Sun and eight
planets plus Pluto. We consider the orbits of the planets as extremely
conservative systems and then we set $R$ and $\bf v$ constant in the
resolution of eq.22, being $\bf v$ the difference between $\bf v_{M}$ the
solar speed, and $\bf v_{m}$ the free motion of the planet. Reading the actual
data from the standard tables nasa , essentially extracting $a$ and $b$,
lengths of major and minor semi axes of the planets orbiting in the solar
system, from that the relative eccentricity $\epsilon$ values, we can obtain
the relative values of $R$ and $\beta$, using the equivalences $R=2a$ and
$\beta=\epsilon$.

Figure 10: The trajectories $(X_{t},Y_{t})$ of the four planets nearest the
Sun are drawn. They have been calculated by eq.22, taking into the account the
equivalences 27-29 and the eq.22. Sun place coincides naturally with the
origin of the axes intersection.

####  IV.3.1 Orbits

Being $D_{t}=\sqrt{X_{t}^{2}+Y_{t}^{2}}$, the couple $(X_{t},Y_{t})$ solving
the eq.22 coincides with the actual orbits described by the nine objects, as
in the fig.10 and fig.11. In this case, the real space $S(x,y)$ and the
virtual one $\Phi(X,Y)$ can be considered almost coinciding probably due to
the fact that $\bf v$, the difference between the solar speed $\bf v_{M}$ and
$\bf v_{m}$, is kept constant in the dynamics of this system. So, the planet
orbits naturally describe an equipotential path, and are really elliptical,
with the Sun, the center of mass, placed at one of the two foci, as provided
by the Kepler’s first law.

Figure 11: The orbits of the four planets more distant from Sun are shown,
plus that of Pluto, the largest one. As in the previous figure, they derive
from the statements 27-29, and solving the eq.22.

####  IV.3.2 Orbital speed

The orbital speed is evaluated for each of the nine bodies belonging to the
solar system, assuming the Sundata  as center of mass. Coming after some
logical steps, the following relation can be adopted for this as well as for
any system with center of mass $M$

$v_{orb}=\sqrt{\frac{2\gamma^{2}GM}{R}\left[1+\beta^{2}-\frac{2\beta
X_{t}}{D_{t}}\right]}.$ (30)

where $G$ is the gravitational constant. The eq.22 is applied along the whole
trajectories, tabulating the average with the minimum and maximum values in
tab.1.

Table 1: Velocity values and orbital period for the planets of the solar system, including Pluto. The formula 30 is applied.  | $v_{(aphelion)}$ | $~{}~{}~{}\bar{v}$ | $v_{(perihelion)}$
---|---|---|---
| $(Km/s)$ | $~{}(Km/s)$ | $~{}(Km/s)$
Mercury | 38.858 | 48.215 | 58.976
Venus | 34.784 | 35.020 | 35.258
Earth | 29.291 | 29.786 | 30.286
Mars | 21.972 | 24.164 | 26.497
Jupiter | 12.440 | 13.063 | 13.705
Saturn | 9.138 | 9.649 | 10.179
Uranus | 6.485 | 6.802 | 7.128
Neptune | 5.385 | 5.432 | 5.478
Pluto | 3.676 | 4.790 | 6.112

These two values correspond to the object positions relatively at aphelion and
perihelion, as expected from Kepler’s second law. The agreement with the
measurements nasa  is good especially for the minimum and maximum values that
regard punctual body positions. Some small discrepancies about the average
value can be due to variability of density of states along the each orbit, and
to the different algorithms used for the elaboration of the average value.

##  V conclusion

Based on euclidean space-time geometry and on analytic procedures, the
quantity $R$ is revealed. It doesn’t correspond simply to geometric distance
between source and detector, rather it is the real signal path covered at $c$
speed. It represents the physics magnitude to take into the account for the
intensity field evaluation, in any $inverse~{}square~{}law$. In light of this
new quantity, classical physical paradigma have been re-examined, bringing out
unsuspected as natural analogies between the geometric and physical parameters
describing the mechanics of moving bodies. Testing the method on a newtonian
system such as the local solar system, the result perfectly fits to the actual
data. The assessment is compatible with relativistic formulas although no
hypothesis was formulated on light special isotropy, neither on the space-time
curvature, posing the issue on the actual need to introduce these concepts in
interpreting the dynamics of moving bodies.

Acknowledgements

I would like to thank my precious teachers and colleagues who inspired this
study, even the many who have allowed to achieve it, with their helpful
suggestions and unfailing encouragements. I am also grateful to Giorgio
Fornetti for his valuable advices and his thorough review of the manuscript.

## References

  * (1) A. Einstein, Annalen der Physik 17, 891-921(1905).
  * (2) A.A. Michelson and E.H. Morley, Am. J. Sci. 34, 333 (1887).
  * (3) T. Roberts $\&$ S.Schleif, ”What is the experimental basis of Special Relativity?”. $math.ucr.edu$ (2007).
  * (4) H.P. Robertson, Rev. of Mod. Phys. 21-3, 378 (1949).
  * (5) Generally, $c$ indicates the speed of light but from the analytical point of view, it represents simply the emission velocity of any signal, without other special physical hypotheses.
  * (6) R.P.Feynman, R.B.Leighton and M.Sands, The Feynman Lectures on Physics Electromagnetism $\bf II$, 21-5, Addison-Wesley, Reading MA (1964).
  * (7) Particle Data Group, Chi. Phys. $\bf C$, Vol. $\bf 38$, 9 (2014).
  * (8) NASA Space Science Data Coordinated Archive, http://nssdc.gsfc.nasa.gov


# Geometric considerations on planetary surface temperatures

Sabin Roman  Centre for the Study of Existential Risk, University of
Cambridge, UK

###### Abstract

We propose a formula for computing the average planetary surface temperatures
based solely on the solar irradiance and the bond albedo. The formula is
empirically derived from data on Earth, Venus and Titan, and a model is
proposed to justify it. We introduce the concept of planetary inner albedo, as
a complement to the usual bond albedo. A geometric proof is given for the main
finding of the paper, which can be summarized as follows: the ratio of the
inner to outer albedo is a constant, related to the universal parabolic
constant. Furthermore, we extend the surface temperature formula to gas
giants, giving the temperature at which condensates (e.g., of ammonia) start
forming within their atmosphere, particularly for Jupiter, Saturn and Uranus.
Based on model complexity, applicability and accuracy, the heating mechanism
via atmospheric reflectivity (a mirror effect) performs much better than the
alternatives. Responses to reviewers are included at the end.

††preprint: APS/123-QED

##  I Introduction

Determining the average surface temperature of Earth-like planets (rocky, with
a significant atmosphere of pressure greater than $10$ kPa) is a fundamental
question in planetary science, and has important implications in particular
for Earth’s biosphere [1, 2]. Black body estimates markedly underestimate the
temperature, not accounting for the full effect of the atmosphere. Calculation
of surface temperatures based on the ideal gas law or the lapse rate can give
accurate estimates, but do not account for the energy balance of incoming and
outgoing radiation [3]. Given the complexity of the atmospheric system,
finding universal regularities can seem unlikely. However, finding such
regularities is essential to our understanding, one example being the linear
decrease of temperature with altitude in the troposphere [4].

In this work we highlight a linear relationship with universal applicability.
Specifically, we plot the ratio of net solar irradiance to emitted radiation
for Titan, Earth and Venus versus their bond albedo. Surprisingly, a linear
fit matches the data very well, see Fig. 1. This implies that the slope is a
constant independent of the features of the celestial bodies, leading us to
propose a geometric rationale for it, which we explore in Section 2. This
provides a way to account for incoming solar radiation and atmospheric
features in determining the surface temperature. It is striking that the
albedo provides sufficient information in this regard. In Section 3, we
discuss how applying the formula to gas giants seems to determine the
temperature at which condensates form in their atmosphere. Condensation forms
key problem related to atmospheric dynamics of the gas giants and the formula
indicates where solar radiation plays the dominant role in determining the
temperature versus other process such as convection. We have included Jupiter,
Saturn and Uranus in Fig. 1 and the linear fit uses the data for rocky bodies
in Table 1.

We can interpret Fig. 1 as follows. Consider the a planet with albedo $\alpha$
and incoming radiation $I$. The amount of radiation passing (net irradiance)
is $I(1-\alpha)$. If the planet radiates at $I_{p}$ with an inner albedo
$\beta$ (fraction of radiation reflected back to the surface), then amount of
radiation getting through is $I_{p}(1-\beta)$. Conservation of energy requires
$I(1-\alpha)=I_{p}(1-\beta)$, and thus:

$\frac{I(1-\alpha)}{I_{p}}=1-r\alpha$ (1)

where we define $r=\beta/\alpha$ as the slope of the linear fit in Fig. 1. We
find that $r\simeq 1.298$.

The surface radiant exitance is calculated using the Stefan-Boltzmann law,
namely $I_{p}=\sigma T_{p}^{4}$ where $T_{p}$ is the surface temperature.
Thus, equation (1) can used to estimate the surface temperature for a given
celestial body, provided it is rocky, with an appreciable atmosphere. If
$\beta=0$, this ignores any atmospheric effect and we obtain the black body
temperature, which underestimates measured values. If $\beta=\alpha$ this
accounts for inner reflection but does not consider radiation coming from a
nearby source. The proximity of the radiation source leads to a geometric
enhancement of the inner albedo, which we detail in the next section. The
temperature estimates for different celestial bodies are listed in Table 1 and
are in a remarkably good agreement with measurements.

The universality of (1) indicates $r$ is geometric in nature, a possibility
which we explore in the next section.

Figure 1: The ratio of incoming to outgoing surface radiation versus the
albedo for Titan, Earth and Venus. Gas giants are included with their
“surface” corresponding to where condensates start forming. Midpoint values
from Table 1 used and $2\%$ error bars for temperature changes included.
Figure 2: Incoming light of irradiance $I_{0}$, passing through the
transparent segments $M$, reflecting off the rough surface $S$ and reflecting
again off the inner surface of $M$. Figure 3: The geometry around a segment
$AB$ of $M$. Lines $AC$ and $BD$ go through the origin, which is the center of
$S$. Tangents at $C$ and $D$ intersect the circle encompassing $AB$ at points
$E$ and $G$. The midpoint of $EG$ is the focus $F$ of the parabola $P$. Light
comes in with irradiance $I_{0}$, reflects of $M$ with exitance $I_{1}$ and
again off $S$ with exitance $I_{2}$.

##  II Model of inner albedo

Given the large differences in atmospheric composition, surface features and
other characteristics between Venus, Earth and Titan, the fact that (1) holds
empirically so well suggests that $r$ is geometric in nature. In this section
we propose a model to account for equation (1), which can justify the
universal nature of $r$.

Consider a reflective sphere $S$ with a rough surface such that lights
scatters in the outward tangent half-space at every point. We make two main
simplifying assumptions, which will be discussed at the end of the section.
The first assumption is that we restrict our analysis to a two dimensional
cross-section made by a plane through the centre of $S$, see Fig. 2. The
cross-sectional circle is surrounded by a set $M$ of circular segments that
are transparent to incoming light but reflective on their inner surface.
Furthermore, the mirrors $M$ are at a close distance from the surface relative
to the radius of the sphere $S$. The incoming light has irradiance $I_{0}$.

We are interested in how the light reflects from the inner surface of $M$, and
we look at a segment $AB$ of $M$ in Fig. 3. The light reflects off $M$ at an
intensity $I_{1}$ and then reflect off the surface $S$ at an intensity
$I_{2}$. The second key assumption is that when the light reflects off $S$ it
does so with a wavefront well-approximated by the parabola $P$ with focus $F$
from Fig. 3.

Let $l_{1}$ be the length of the circle segment $AB$ and $L_{1}$ be the length
of the parabolic segment $EG$. The following equality holds between the
incoming light $I_{0}+I_{1}$ (radiation passing through $M$ plus the
reflected) and reflected off the surface with intensity $I_{2}$:

$l_{1}(I_{1}+I_{0})=L_{1}I_{2}$ (2)

This express conservation of energy. In addition, we expect $I_{0}=I_{2}$
because $S$ cannot emit more than it receives. Hence:

$I_{1}=\left(\frac{L_{1}}{l_{1}}-1\right)I_{0}$ (3)

From the geometry in Fig. 2 we can deduce the equality $FG\simeq AB\simeq
l_{1}$, see Appendix A. The ratio of the parabolic segment $EG$ of length
$L_{1}$ to the linear segment $FG\simeq l_{1}$ is given by the universal
parabolic constant $p=\log{(1+\sqrt{2})}+\sqrt{2}\simeq 2.2956$. Then, the
ratio between the reflected light from $M$ and the outgoing light from $S$ is:

$\frac{I_{1}\sum_{i}l_{i}}{I_{0}C}=(p-1)\alpha$ (4)

where $\sum_{i}l_{i}$ is the summed lengths of all the segments of $M$, which
make up a fraction $\alpha$ of the circumference $C$ of the circle. Thus, we
can consider the inner albedo of $M$ to be $\beta=r\alpha$ where $r=p-1\simeq
1.2956$. This theoretical value gives a close match of temperature predictions
with data, see Table 1.

The above analysis made two key simplifying assumptions: (1) a restriction to
a two-dimensional setting, and (2) the wavefront of the light emitted from the
surface $S$ is parabolic. Regarding assumption (2), we see that the parabola
$P$ approximates a circular wavefront, which is represented by a grey curve in
Fig. 3. Thus, Huygens’ principle is not violated. In particular, the grey
circle’s radius is 5/2 times the focal length of the parabola.

Table 1:  Predictions of surface temperatures using equation (1) and the theoretical value of $r=\log{(1+\sqrt{2})}+\sqrt{2}-1$. References in table, otherwise data is from [5]. Planet | Solar irr. (W/m${}^{2})$ | Albedo |  $T_{p}$(K) | $T(K)$
---|---|---|---|---
Venus | 650.3 | 0.765 [6, 5] | 737 | 742.3
Earth | 340.2 | 0.306 | 288.15 | 288.2
Titan | 3.7 | 0.260 [7] | 90.6-94 [8, 9] | 92.4
Jupiter | 12.6 | 0.503 [10] | 132.79 [11] | 133.3
Saturn | 3.7 | 0.342 [12] | 93.5 [13] | 93.7
Uranus | 0.923 | 0.300 [5] | 64.4-66.6 [14] | 65.7

Regarding assumption (1), we can generalize the argument to three dimensions.
A first intuition would be to consider radially symmetric generalizations,
such as the light having a paraboloidal wavefront. However, this is not
justified because the reflective segments $M$ are irregular (e.g., the shape
of clouds) and not necessarily circular disks. The two dimensional restriction
in Fig. 2 does qualitatively capture a cross section of a planet with an
atmosphere. This can be generalized by considering cross-sectional thin slabs
wherein the wavefront consists of short parabolic cylinders. This implies a
translational symmetry over short distances and keeps the proportions and
above calculations unchanged. Huygens’ principle is again obeyed in virtue of
that fact that a short parabolic cylinder approximates a thin cross-section of
a sphere.

The argument we have presented is geometric in nature and applies to rocky
celestial bodies such as Venus, Earth and Titan. In the next section we
discuss extensions to gas giants.

##  III Discussion and applications

The physical interpretation of equation (1) is relatively straightforward. The
solar radiation that reaches the ground heats it up. The ground then emits
thermal radiation that gets reflected by the clouds. The reflected radiation
adds to the heating of the ground that will then radiate more as a
consequence. The ground keeps heating up until the net radiation getting
through to outer space matched the incoming radiation, as (1) states. As we
have show, this physical mechanism is enough to account for the surface
heating of celestial bodies in the solar system with a notable atmosphere.

Because the inner albedo is proportional to the planetary bond albedo, and the
constant of proportionally is geometric in nature, this indicates that the
physical processes determining the inner albedo are the same as the bond
albedo. For this to happen it is sufficient that the bottom-side of clouds has
the same reflective properties as their top-side, which is a reasonable
assumption. The spectrum of incident light is different for the top and
bottom-side of the clouds but if the reflection is uniform across all
wavelengths then the planetary albedo $\alpha$ and inner albedo $\beta$ are
determined by the same underlying physical processes. Furthermore, the model
encapsulated in equation (1) is arguably the simplest possible model that
could achieve such a result, as it accounts for the two main influences on the
temperature (the Sun and the atmosphere) via one parameter each, for a total
of two measurable parameters.

Table 1 uses the solar irradiance from NASA [5] (given by the solar constant
of a given body divided by 4). There are some sensitivity issues with (1) that
require discussion. For Venus’ albedo we use the average of the two quotes
values in the literature, 0.76 [6] and 0.77 from the NASA website [5]. These
two albedo values yield a temperature interval of $650-1024$K which indicates
an increased sensitivity of (1) in certain regions of parameter space.
However, this can also be seen as a high-precision predictive tool giving a
testable output. For Earth and Titan, a change of $10\%$ in albedo gives
temperature ranges of $286.7-289.9$K and $92-92.7$K, respectively. So, in
these latter cases the albedo sensitivity issue does not arise.

The formation of condensates in the gas giants has long been a matter of
debate in the scientific community [15, 16, 17]. Specifically, in the case of
Jupiter, explaining the lack of spectral evidence of ammonia posed a challenge
[18, 19]. However, a number of mechanisms have been put forth [19, 20, 16, 17,
21] which have advanced our understanding. Nevertheless, the atmospheric
structure and dynamics of the giant planets retains many open problems [22].

While we not provide detailed causal mechanisms, we do highlight the fact that
equation (1) does give temperature values for the gas giants that seem to
correspond to the layer in the atmosphere where condensates start to form. The
temperature estimate for Jupiter is consistent with the temperature of
$132.79$K at 0.5 bar [11, Table 7], above which NH3 saturation would occur,
leading to ammonia clouds. Similarly, for Saturn there’s a drop in the lapse
rate between 0.5 and 0.3 bar which indicates a transition to a lower
convective haze layer [13]. At $0.275$ bar, the temperature is $93.5$K [13,
Table 1], again consistent with our result for Saturn. For Uranus,
condensation of ethane occurs at $14$ mbar with a atmospheric temperature of
$64.4$K and evaporation is at $600$ mbar with a temperature of $66.6$ [23]. We
quote this temperature range in Table 1 and it is consistent with our
prediction. Application of equation (1) to Neptune yields $52.4$K which is
close to the tropopause temperature of $53$K [24] but the relation to
condensates is uncertain in this case.

We can interpret formula (1) as giving the temperature determined by solar
radiation within the atmosphere of gas giants. The temperature estimate thus
lies at the boundary where convection start to become a main heat transfer
mechanism, with condensation being an indicator of this.

The estimates for the surface temperature of Titan range in the interval
$90.6-94$K [8, 9]. It is interesting to note that the same temperatures are
also found at higher altitudes, where a condensate haze forms at a pressure of
$0.03$ bar [25]. While it is not obvious why this should be case, it is
nevertheless consistent with our findings for the gas giants.

We note that for Jupiter a long-time recorded value of the bond albedo was
0.343 [5], still quoted on the NASA website at the time of this writing. The
bond albedo value was revised to approximately 0.5 [10], showing that
planetary data can be subject to substantial, large corrections. The albedo
values for Titan has also been updated recently, from 0.22 [26] to a seasonal
variation between 0.252-0.265 [7]. Thus, the predictions of (1) are compatible
with current knowledge, and well within the uncertainty of measurements and
estimates for the temperature and albedo in the literature.

Other bodies in the solar system, such as Mercury, the Moon, Mars and Triton,
do not have a substantial atmosphere and hence there is little to no cloud
cover. Furthermore, their surface temperature fluctuates significantly [27,
28, 29, 30]. This makes defining an average temperature less meaningful and
not suitable to applying equation (1).

##  IV Conclusion

We have presented the empirical equation (1) that allows us to determine
surface temperatures of rocky celestial bodies with substantial atmospheres.
Remarkably, only the solar irradiance and bond albedo are sufficient to
determine the surface temperature. We have provide a geometric justification
for this fact, where the universal parabolic constant pays a key role in
determining the ratio between inner and outer bond albedo. In addition, we
have extended the applicability of equation (1) to gas giants, finding the
temperature where condensates start to form. We hope the work inspires future
studies into these matters.

In terms of comparison with other theories, we only note that early work by
Hansen [31] reduces a climate model to an empirical formula with 11 fined
tuned parameters to account for a temperature change of at most a few degrees
on Earth’s surface. More recent models tend to be even more complex [32]. The
proposed mechanism in this paper is based on cloud reflectivity and
conservation of energy, using only 2 measurable parameters to account for the
entire temperature at the surface of all celestial bodies in the solar system
with a notable atmosphere. Hence, based on model complexity (number of
parameters and equations), applicability and accuracy, the heating mechanism
we propose performs much better than the alternatives in accounting for and
explaining measured temperatures.

## Disclaimer

The ideas, theories and interpretations expressed in all past and present
versions of this paper are solely of the author and do not express the views
or opinions of my employer nor of my collaborators.

## Appendix A

The focus $F$ has $x_{F}=0$. For coordinate $x_{G}$ we have:

$\displaystyle x_{G}$
$\displaystyle=\frac{R}{R+h}x_{B}+\frac{y_{B}}{R+h}\sqrt{x_{B}^{2}+y_{B}^{2}-R^{2}}$
(5) $\displaystyle\simeq 2x_{B}$

where $R$ is the radius of $S$, $h$ is distance between $S$ and $M$. The
coordinate $x_{A}=-x_{B}$, so $AB=2x_{B}$. We assume $h<<AB<<R$, then the
length of the circular segment is $l_{1}\simeq AB\simeq FG$.

## Appendix B

Reviewer comments in italics. The reviews are of a submitted version of the
article that closely matches the first arXiv version. The journal is well-
establish in the field of planetary science. The reviewers are referring to
Table 2 in Appendix B, which uses some older data than Table 1.

Table 2:  Estimates of $r$ from equation (1). References in table, otherwise data is from [5]. Average $r=1.295$. Planet | Solar irr. (W/m${}^{2})$ | Albedo |  $T_{p}$(K) | $r$
---|---|---|---|---
Venus | 650.3 | 0.760 [6] | 737 | 1.303
Earth | 340.2 | 0.306 | 288 | 1.290
Titan | 3.7 | 0.220 [26] | 92 [8, 9] | 1.316
Jupiter | 12.6 | 0.503 [10] | 132.79 [11] | 1.284
Saturn | 3.7 | 0.342 [12] | 93.5 [13] | 1.281

Reviewer 1

My first reason for suggesting rejection is the physical basis for the formula
proposed. The model set up in Equation 1 balances the incoming solar
radiation, reduced by reflection from the Bond albedo, against the radiation
emitted from the surface, reduced by an “inner albedo” representing
scattering. This omits the centrally important process of absorption and re-
emission by the atmosphere. This does not act like a mirror reflecting light
back to the surface and allowing some to pass through, as is proposed in this
model. A typical energy balance model would balance (at the top of the
atmosphere) incoming solar radiation, outgoing radiation from the surface that
has not been absorbed by the atmosphere, and outgoing radiation emitted
directly from the atmosphere. This key process is omitted from the model.

How successful has the theory based on absorption and re-emission processes in
the atmosphere in accounting for surface temperatures? A specific model
comparison would be needed in terms of model complexity (number of parameters,
equations etc.), accuracy and overall connection to other established, proven
mechanisms. I do not necessarily see any reason why the processes of
absorption and re-emission would be precluded from the mechanism proposed in
this paper provided that: (1) they fill in an explanatory gap in determining
surface temperature and (2) that they would occur at a higher level of
granularity in modelling the physical systems at play. However, regarding (1),
there is little to no explanatory gap left between the predictions of the
model proposed in this paper and the observed temperatures. Regarding (2), a
more detailed model should not impact the physical predictions at a coarser
scale, but would have to be consistent with them. Given the potentially large
literature on the topic of atmospheric absorption and re-emission the reviewer
did not pinpoint any specific literature or model. This is unfortunate as
concrete examples would serve as a more constructive feedback and give the
chance for a comparison of specific theories. We have included a comparison
with work by Hansen [31] in the current (updated) version of the paper, see
the Conclusion section.

However, I have tried to keep an open mind that the model proposed could be an
idealised representation than somehow includes atmospheric emission, so
assessed the empirical results based on Figure 1 on their own merits. The
match of Equation 1 to real data (for constant r) appears to be an
extraordinary result, as it matches two physically separate processes. The
Bond albedo alpha relates to reflection of mostly visible light with short
wavelengths from the Sun, by features like clouds or reflective surfaces,
while the “inner albedo” beta must correspond to the interaction (absorption
or scattering, by gases like CO2) of the atmosphere with longer infrared
wavelengths emitted by the surface. These two processes have no intrinsic
relation - is it perfectly plausible that the Earth could have a more
reflective surface (e.g. more clouds or desert coverage) but the same amount
of CO2 in its atmosphere. If Equation 1 holds for constant r, it would
therefore show a very surprising result about two apparently independent
processes.

The reviewer is incorrect is assuming that the “inner albedo” beta must
correspond to the interaction (absorption or scattering, by gases like CO2) of
the atmosphere with longer infrared wavelengths emitted by the surface. We
write that: “Because the inner albedo is proportional to the planetary bond
albedo, and the constant of proportionally is geometric in nature, this
indicates that the physical processes determining the inner albedo are the
same as the bond albedo. For this to happen it is sufficient that the bottom-
side of clouds has the same reflective properties as their top-side, which is
a reasonable assumption. The spectrum of incident light is different for the
top and bottom-side of the clouds but if the reflection is uniform across all
wavelengths then the planetary albedo $\alpha$ and inner albedo $\beta$ are
determined by the same underlying physical processes.”

My second reason for suggesting rejection is therefore the analysis of the
data in Figure 1. This initially appears to be a remarkable match between five
separate data points. However, I do not believe that this is the case, and
will discuss each in turn. First, Jupiter and Saturn are essentially
retrofitted to the existing trend of r=1.3, deriving a Tp value that matches
this and then justifying it post-hoc as similar to the temperature at which
condensates form, although it is unclear as to why this is then analogous to a
surface. Second, if Equation 1 is rearrange to express r in terms of the other
variables, it becomes clear that the value of r 1.3 for Venus essentially only
depends on its albedo (r=1/alpha), as the surface temperature is so high that
it makes the term it is in negligible. This casts doubt on this as a general
link between the two processes - r would equal 1.3 if the surface temperature
were 600K, 700K, or 800K, and the albedo of Venus could easily be different.

Observations of Earth indicate that the albedo is surprisingly constant over
time [33]. The albedo anomaly oscillates up to $0.2-0.4\%$. There’s no
particular reason why this robustness in the albedo’s average value would be
characteristic to Earth. A seasonal variation of up to $5\%$ in the albedo
value has been measured for Titan [7], however this has little effect on our
temperature estimate. So, it’s reasonable to assume that similar properties of
the bond albedo (small oscillations around a stable central value) would hold
for the other celestial bodies. Hence, the bond albedo of Venus could not “be
easily different”. The reviewer is correct in stating that the $r$ parameter
is approximately the inverse of Venus’ albedo but this is not exact. Their
product is $0.99$, meaning $1\%$ of the radiation from the surface of Venus
escapes. Hence, a different value of $r$ would have a marked effect in
determining the temperature on Venus. However, the way the reviewer framed his
reasoning about $r$ is incorrect. The solar irradiance and albedo determine
the surface temperature temperature, and $T_{p}$ is high because of the
specific values of those parameters. The regression in Fig. 1 is meant to
suggest the constancy of $r$, while the validation comes in calculating the
surface temperatures and seeing the good agreement with measurements by taking
a specific, constant, theoretical justified value for $r$. This is done in the
new version of the paper in Table 1.

For Titan, the value of r is instead essentially entirely determined by the
surface temperature, and only the specific value of 92K chosen gives 1.3. The
McKay 1991 paper cited gives the surface temperature of Titan at 94K, which
would give r=1.58. The Jennings 2016 paper cited shows variation in the range
90 to 94K but do not actually quote the mean value as 92K, and visually from
their graphs it appears to be higher than 92K. They also quote a standard
deviation on their measurements of 0.4K - calculating r in the range 91.6K to
92.4K gives a range of r from 1.25 to 1.38.

As stated in the abstract, the goal of the paper is the calculation of surface
temperatures. The estimates of $r$ are given to suggest a certain numerical
value that has to be justified theoretically. In Section 2 and Table 1, we
have done precisely this. The fact that the reviewer is trying to argue that
$r$ is not constant or not well determined for Titan (or other cases) is
missing the point. We can discard Fig. 1 and Table 2 altogether as long as we
have obtained a candidate value of $r$ and have found a precise mathematical
constant coming from a theoretical argument. For Titan, the model predict
$92.4$K whereas the estimates from the literature are $90.6-94$K which is a
reasonable agreement. At worst, the model accounts for $98\%$ of the
temperature on Titan but, given the uncertainties in the literature values,
the explanatory gap might not exist at all.

The Earth value of roughly r=1.3 does appear to be consistent, but I do not
believe any of the other data points to be robust for the reasons above; it
would have been vital to attach error bars to the plotted points to find some
of the above issues. In general, the downward trend in Figure 1 relates to the
fact that atmospheres with higher albedo tend to have higher atmospheric
absorption and scattering of thermal radiation from the surface, as they are
thicker and contain more optically active species. It is not surprising if a
couple of points on this general trend can be joined together with an apparent
linear trend. I also believe that the case of Mars was dismissed too readily;
this is still an atmosphere with clouds and a greenhouse effect, which gives a
very different value of r when calculated.

The best available measurements we have are for Earth and so this data
provides the most stringent test for the model. The prediction is $288.2$K for
a surface temperature of $288.15$, an excellent agreement and the best of the
cases we have analysed. As the reviewer acknowledges, the results for Earth
are consistent. The rest of the comment derives from a misunderstanding of the
physical basis of the proposed model, as we outlined in a previous reply. The
reviewer is dismissing a marked regularity in the available data of planetary
bodies, rationalizing and trivializing it based on a misunderstanding. The
inner albedo is not determined by what the reviewer has listed, and with a
thin atmosphere less the $1\%$ of Earth’s pressure the average surface
temperature of Mars is close to the black-body estimate. Furthermore, clouds
on Mars are rare and this prohibits the mechanism we proposed to heat up the
surface. Interesting, the abundance of CO2 does little to nothing to shift the
average temperature from the black-body estimate.

I hope it is clear that I have tried to treat this paper on its own merits
despite its non-standard approach. I must recommend rejection as both the
theoretical model and data analysis have major issues.

We disagree with the reviewer that the attempt of understanding the paper was
not on its merits, but rather bias from pre-existing positions. Firstly, the
reviewer failed to see that inner reflection of radiation (that determines the
inner albedo) proceeds by the same mechanisms through which sunlight is
reflected off the top of the atmosphere (and determines the bond albedo).
Secondly, the reviewed failed at recognizing the fact that varying estimates
of $r$ do not impinge on the validity of the temperature predictions (a
single, theoretical justified value of $r$ that gives accurate predictions is
sufficient for our arguments to hold). Thirdly, the reviewer incorrectly
assumes that the planetary albedo can easily vary, which indicates a lack of
knowledge or expertise in the field. Fourthly, the reviewer disregards a
remarkable regularity in the solar system that does stand out from the data,
despite the uncertainties in the measurements. We would not like to imagine
how science would have progressed if every linear relationship discovered was
deemed and dismissed as a “couple of points on this general trend can be
joined together with an apparent linear trend.” There are cases of spurious
correlation, however we are looking at incoming and outgoing thermal
radiation, two causally connected quantities.

Reviewer 2

This manuscript presents an interesting theoretical work estimating the
temperature of celestial bodies with significant global-scale atmospheres. The
most important result is equation (1) presented in the manuscript, which
establishes a relationship between the input radiation (i.e., the absorbed
solar radiation) to a celestial body and the radiation radiated by the
celestial body. The input radiation is determined by the incoming solar
radiation and the Bond albedo (alpha). The incoming solar radiation at a given
celestial body is further determined by the distance between the Sun and the
celestial body. Therefore, the Bond albedo is the only parameter controlling
the input radiation. On the other hand, the output radiation (i.e., the
radiation a celestial body radiates) is expressed as the blackbody radiation
at the surface (or a corresponding pressure level) multiplied by a factor less
than 1. The factor is further expressed as a function of a new parameter of
inner albedo (beta), 1-beta. The author further suggests that the ratio
between the Bond albedo (alpha) and the inner albedo (beta) is a constant for
these celestial bodies with significant atmospheres. Therefore, equation (1)
can be used to determine the temperature of these celestial bodies with
significant atmospheres with known Bond albedos. The author also presented a
geometric proof and brief discussion on its applications. The topic and idea
are interesting, but I have major concerns as follows.

The reviewer has accurately summarized the paper.

The author claimed a constant parameter for the ratio between the Bond albedo
and the inner albedo (i.e., r) for these astronomical bodies with significant
atmospheres. However, such a constant should be carefully examined. For Earth
and Titan, there are updated values for the Bond albedo (e.g., Loeb et al.,
2020; Creecy et al., 2021). I expect that the updates of Bond albedo and solar
constant at Titan and Earth do not significantly affect the value of the ratio
r. But it is still preferred to use more precise values. For Venus, the
global-average Bond albedo surface temperature has large uncertainties due to
limitations in the current available observations (e.g., Tomasko et al., 1980;
Titov et al., 2013). In addition, it does not make sense that Venus’ surface
radiation plays any significant role in its radiation budget (equation (1))
considering its thick atmosphere. The radiation from the surface will be
completely blocked by its thick atmosphere before it escapes into space.

According to our estimates the surface radiation of Venus is not completely
blocked, but is $99\%$ blocked. The $1\%$ escaping energy must match the
amount of incoming radiation, which implies the surface temperatures have to
be very high. We thank the reviewer for the references. The new albedo value
for Titan does not significantly alter the resulting temperature.

The most serious trouble for equation (1) comes from giant planets. First,
Uranus and Neptune have a similar temperature profile of the troposphere and
Bond albedo, but the solar constant (i.e., the incoming radiation) is
significantly different between the two ice giants because of the different
distances from the Sun for the two planets. It is hard to believe that they
follow equation (1) with a constant (i.e., r = 1.3).

In the first version of the paper we initially dismissed Uranus and Neptune
due to a not significant deviation of our prediction and the usual black-body
estimate. It is peculiar that the reviewer mentions them. In the current
version, we have found for Uranus similar results to Saturn and Jupiter, with
condensation and evaporation of ethane occurring at the temperature predicted
by our model.

Second, the study by Hanel et al. (1983) seems to underestimate the Bond
albedo of Saturn (personal communication with a group of scientists who are
re-examining Saturn’s radiant energy budget with the Cassini long-term multi-
instrument observations).

An explicit value of the albedo would be appreciated. However, even a 20%
change doesn’t alter the temperature prediction for Saturn significantly, with
the predicted temperature range being $92.5-95$K.

Finally, the energy budget is more complicated for giant planets than for
terrestrial bodies. Unlike terrestrial bodies, internal heat plays a critical
role in the energy budget of the atmospheric system for giant planets.
Jupiter, Saturn, and Neptune all have confirmed internal heat, and it plays a
critical role in the energy budgets of the three giant planets. But the
internal heat is not considered in equation (1).

Indeed, the internal heat is not considered as the formula determines the
temperature due to radiation at the boundary where condensation starts and
convection becomes important.

The physics of the new parameter (i.e., inner albedo) suggested by the author
is unclear. The radiation budget is determined by the absorbed solar radiation
(i.e., the left-hand expression of equation (1)) and the emitted thermal
radiation. The emitted thermal radiation is equivalent to the right-hand
expression of equation (1), but the emitted thermal radiation has much clearer
physics than the right-hand expression including the inner albedo. For Titan
and Earth, both radiations from the solid surface and atmosphere (mainly the
troposphere) contribute to the total emitted radiation. The temperature
decreases with height in the troposphere for both Titan and Earth, so the
surface’s radiation is stronger than the atmospheric radiation. That is the
reason why there is a factor smaller than 1 (i.e., 1-beta) before the term of
the surface’s radiation on the right-hand side of equation (1) if we use the
surface’s blackbody radiation (i.e., $Ip=sigma*Tp^{4}$) to approximate the
radiation radiated from the two terrestrial bodies (i.e., Titan and Earth).
For Venus, I do not think that the solid surface plays any significant role in
the radiation budget, as I discussed as above.

We have now explained the physical basis of the proposed model in the
Discussion section and in a reply to another reviewer comment. Given the
empirical success of the proposed model, see Table 1, there is potentially
extensive rethinking necessary for a good deal of literature on planetary
energy balances. It is not within the scope of this article to cover ever
possible consequence of the ideas we introduce.

For Jupiter and Saturn, the pressure levels that dominantly contribute to the
emitted energy are around 200-300 mbar (Li et al., 2010, 2012), which is above
the top visible clouds in the upper troposphere around 400-700 mbar (e.g.,
Banfield et al., 1998; Perez-Hoyos and Sanchez-Lavega, 2006). Thus, there will
be also a factor less than 1 if the author assumes the blackbody radiation
from a level at 500 mbar and 300 mbar for Jupiter and Saturn, respectively. In
summary, a factor less than 1 appearing on the right-hand side of equation (1)
can be explained and expected from the fact that the atmospheric layers above
the solid surface for some terrestrial bodies or the pressure levels above the
top visible clouds for gas giants contribute to the emitted thermal energy. I
do not see a clear physics for the new parameter - inner albedo (i.e., beta)
suggested by the author.

We have expanded our explanation of what determines the inner albedo (see
Discussion section). Reflection of radiation from the bottom-side of clouds
(present down to approximately $0.5-0.6$ bars) is the main mechanism
underlying it. Planets with no to little cloud cover, such as Mars, will not
have heating generated by the proposed mechanism and so will have average
surface temperatures close to the black-body estimate.

The geometrical proof needs more detailed explanations. First, some
assumptions need to be validated. For example, the assumption of simplifying
the scattering radiance to a simple geometry does not seem to be reasonable
considering the complicated scattering properties of the surface and
atmosphere (e.g., Hapke, 2012). Second, more details should be provided. For
example, Equations (2) and (3) need more explanations. It is hard to tell if
the geometric proof is correct with the lack of details.

Indeed, there is further work that can be done to understand the origins of
the $r$ parameter (or constant). We have only aimed at giving a first
indication of the possible origin of $r$, based on a geometrical rationale.
Given the potential breakthrough advances in the current paper, we consider
the rest of the scientific community with more resources at their disposal (in
time, finances and personnel) can continue the work to understand our findings
and address the reviewer’s concerns.

## References

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