Vol.2, No.4, 329-337 (2010) Natural Science
Copyright © 2010 SciRes. OPEN ACCESS
Orbital effects of Suns mass loss and the Earths fate
Lorenzo Iorio
INFN-Sezione di Pisa, Viale Unità di Italia 68, Bari (BA), Italy; Ulorenzo.iorio@libero.itU
Received 22 November 2009; revised 28 November 2009; accepted 13 January 2010.
I calculate the classical effects induced by an
isotropic mass loss
of a body on the or-
bital motion of a test particle around it; the
present analysis is also valid for a variation
of the Newtonian constant of gravitation. I per-
turbatively obtain negative secular rates for the
osculating semimajor axis a, the eccentricity e
and the mean anomaly , while the argument
of pericenter ω does not undergo secular pre-
cession, like the longitude of the ascending
node Ω and the inclination I. The anomalistic
period is different from the Keplerian one, being
larger than it. The true orbit, instead, expands,
as shown by a numerical integration of the eq-
uations of motion in Cartesian coordinates; in
fact, this is in agreement with the seemingly
counter-intuitive decreasing of a and e because
they only refer to the osculating Keplerian el-
lipses which approximate the trajectory at each
instant. By assuming for the Sun
 , it turns out that the Earth's perihe-
lion position is displaced outward by 1.3 cm
along the fixed line of apsides after each revo-
lution. By applying our results to the phase in
which the radius of the Sun, already moved to
the Red Giant Branch of the Hertzsprung-Russell
Diagram, will become as large as 1.20 AU in
about 1 Myr, I find that the Earth's perihelion
position on the fixed line of the apsides will in-
crease by .. AU (for
 ); other researchers point towards an
increase of .. AU. Mercury will be
destroyed already at the end of the Main Se-
quence, while Venus should be engulfed in the
initial phase of the Red Giant Branch phase; the
orbits of the outer planets will increase by
.. AU. Simultaneous long-term numeri-
cal integrations of the equations of motion of all
the major bodies of the solar system, with the
inclusion of a mass-loss term in the dynamical
force models as well, are required to check if the
mutual N-body interactions may substantially
change the picture analytically outlined here,
especially in the Red Giant Branch phase in
which Mercury and Venus may be removed from
the integration.
Keywords: Gravitation; Stars; Mass-Loss; Celestial
I deal with the topic of determining the classical orbital
effects induced by an isotropic variation
of the
mass of a central body on the motion of a test particle;
my analysis is also valid for a change
of the
Newtonian constant of gravitation. This problem, al-
though interesting in itself, is not only an academic one
because of the relevance that it may have on the ultimate
destiny of planetary companions in many stellar systems
in which the host star experiences a mass loss, like our
Sun [1]. With respect to this aspect, my analysis may be
helpful in driving future researches towards the imple-
mentation of long-term N-body simulations including
the temporal change of GM as well, especially over
timescales covering paleoclimate changes, up to the Red
Giant Branch (RGB) phase in which some of the inner
planets should be engulfed by the expanding Sun.
Another problem, linked to the one investigated here,
which has recently received attention, is the observa-
tionally determined secular variation of the Astronomical
Unit [2-5]. Moreover, increasing accuracy in astrometry
pointing towards microarcsecond level [6], and long-
term stability in clocks [7] requires to consider the pos-
sibility that smaller and subtler perturbations will be
soon detectable in the solar system. Also future planetary
ephemerides should take into account .
phenomena which may show connections with the prob-
lem treated here are the secular decrease of the semima-
jor axes of the LAGEOS satellites, amounting to 1.1 mm
d1 [8], and the increase of the lunar orbits eccentricity
[9]. However, a detailed analysis of all such issues is
beyond the scope of this paper.
330 L. Iorio / Natural Science 2 (2010) 329-337
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Many treatments of the mass loss-driven orbital dy-
namics in the framework of the Newtonian mechanics,
based on different approaches and laws of variation of
the central bodys mass, can be found in literature
[2,4,10-18] and references therein.
The plan of the paper is as follows. Section 2 is de-
voted to a theoretical description of the phenomenon in a
two-body scenario. By working in the Newtonian frame-
work, I will analytically work out the changes after one
orbital revolution experienced by all the Keplerian orbit-
al elements of a test particle moving in the gravitational
field of a central mass experiencing a variation of its GM
linear in time. Then, I will clarify the meaning of the
results obtained by performing a numerical integration
of the equations of motion in order to visualize the true
trajectory followed by the planet. Concerning the me-
thod adopted, I will use the Gauss perturbation equations
[19,20], which are valid for generic disturbing accelera-
tions depending on position, velocity and time, the
“standard” Keplerian orbital elements (the Type I acc-
ording to [16]) with the eccentric anomaly E as “fast”
angular variable. Other approaches and angular variables
like, e.g. the Lagrange perturbation equations [19,20],
the Type II orbital elements [16] and the mean anomaly
could be used, but, in my opinion, at a price of major
conceptual and computational difficulties1. With respect
to possible connections with realistic situations, it should
be noted that, after all, the Type I orbital elements are
usually determined or improved in standard data reduc-
tion analyses of the motion of planets and (natural and
artificial) satellites. Instead, my approach should, hope-
fully, appear more transparent and easy to interpret, al-
though, at first sight, some counter-intuitive results con-
cerning the semimajor axis and the eccentricity will be
obtained; moreover, for the chosen time variation of the
mass of the primary, no approximations are used in the
calculations which are quite straightforward. However, it
is important to stress that such allegedly puzzling fea-
tures are only seemingly paradoxical because they will
turn out to be in agreement with numerical integrations
of the equations of motion, as explicitly shown by the
Figures depicted. Anyway, the interested reader is ad-
vised to look also at [16] for a different approach. In
Section 3, I will apply our results to the future Sun-Earth
scenario and to the other planets of the solar system.
Section 4 summarizes my results.
By defining  at a given epoch 0, the accelera-
tion of a test particle orbiting a central body experienc-
ing a variation of is, to first order in 0,
= ()
21 +
, (1)
with  |=0.  will be assumed constant
throughout the temporal interval of interest = 
0, as it is the case, e.g., for most of the remaining life-
time of the Sun as a Main Sequence (MS) star [1]. Note
that  can, in principle, be due to a variation of both the
Newtonian gravitational constant G and the mass M of
the central body, so that
. (2)
Moreover, while the orbital angular momentum is
conserved, this does not happen for the energy.
By limiting ourselves to realistic astronomical scena-
rios like our solar system, it is quite realistic to assume
(0)1 (3)
over most of its remaining lifetime: indeed, since
is of the order of 2 1014 yr 1, for the Sun [1], the con-
dition (3) is satisfied for the remaining2 7.58 Gyr
before the Sun will approach the RGB tip in the
Hertzsprung-Russell Diagram (HRD). Thus, I can treat it
perturbatively with the standard methods of celestial
The unperturbed Keplerian ellipse at epoch 0, as-
sumed coinciding with the time of the passage at perihe-
lion , is characterized by
=(1cos ),
cos =cos 
1cos ,
sin =12sin
where a and e are the semimajor axis and the eccentricity,
respectively, which fix the size and the shape of the un-
changing Keplerian orbit,  3
is its unper-
turbed Keplerian mean motion, f is the true anomaly,
reckoned from the pericentre, and E is the eccentric
anomaly. Eq.4 characterizes the path followed by the
particle for any > if the mass loss would sudden-
ly cease at . Instead, the true path will be, in general,
different from a closed ellipse because of the perturba-
tion induced by , and the orbital parameters of the os-
culating ellipses approximating the real trajectory at
each instant of time will slowly change in time.
Think, e.g.
, about the cumbersome expansions in terms of the mean
anomaly and the Hansen coefficients, the subtleties concerning the
choice of the independent variable in the Lagrange equations for the
semimajor axis and the eccentricity [19].
About 80% of such a mass-
loss is due to the core nuclear burning,
while the remaining
20% is due to average solar wind.
The age of the present-day MS Sun is 4.58 Gyr, counted from its z
-age MS star model [1].
L. Iorio / Natural Science 2 (2010) 329-337 331
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2.1. The Semimajor Axis and the Eccentricity
The Gauss equation for the variation of the semimajor
axis a is [19,20]
 =2
12sin +
, (5)
where and are the radial and transverse, i.e.
orthogonal to the direction of
, components, respec-
tively, of the disturbing acceleration, and (12)
is the semilatus rectum. In the present case
2, (6)
i.e. there is an entirely radial perturbing acceleration. For
 <0, i.e. a decrease in the bodys GM, the total gravi-
tational attraction felt by the test particle, given by (1), is
reduced with respect to the epoch .
In order to have the rate of the semimajor axis aver-
aged over one (Keplerian) orbital revolution (6) must be
inserted into (5), evaluated onto the unperturbed Keple-
rian ellipse with (4) and finally integrated over  2
from 0 to 2 because 2
(see below).
Note that, from (4), it can be obtained
. (7)
As a result, I have 4
(sin )sin
0 (8)
Note that if decreases, a gets reduced as well:
 <0. This may be seemingly bizarre and coun-
ter-intuitive, but, as it will be shown later, it is not in
contrast with the true orbital motion.
The Gauss equation for the variation of the eccentric-
ity is [19,20]
 =12
 sin +cos +1
 (9)
For = it reduces to
 =12
2 
 , (10 )
so that 
=(1 + )
; (11)
also the eccentricity gets smaller for  <0.
As a consequence of the found variations of the oscu-
lating semimajor axis and the eccentricity, the osculating
orbital angular momentum per unit mass, defined by
2(12), remains constant: indeed, by using (8)
and (11), it turns out
=(12)2=0. (12)
The osculating total energy 2
according to 
(1) . (13)
Moreover, the osculating Keplerian period
Kep 23
, (14)
which, by definition, yields the time elapsed between
two consecutive perihelion crossings in absence of per-
turbation, i.e. it is the time required to describe a fixed
osculating Keplerian ellipse, decreases according to
 = 3
. (15)
As I will show, also such a result is not in contrast
with the genuine orbital evolution.
2.2. The Pericenter, the Node and the
The Gauss equation for the variation of the argument of
pericentre ω is [19,20]
 =12
 cos+1 +
cos 
 , (16)
where I and Ω are the inclination and the longitude of
the ascending node, respectively, which fix the orienta-
tion of the osculating ellipse in the inertial space. Since
 
and  
depend on the normal component
of the disturbing acceleration, which is absent in the
present case, and =, I have
(sin )(cos)
0= 0. (17)
The osculating ellipse does not change its orientation
in the orbital plane, which, incidentally, remains fixed in
the inertial space because =0 and, thus,  
 
= 0.
2.3. The Mean Anomaly
The Gauss equation for the mean anomaly , defined
as , [19,20] is
 = 2
 +cos 
. (18 )
It turns out that, since
()3(sin ), (19)
4Recall that the integration is taken over the unperturbed Keplerian
ellipse: that is why
a and e
are kept out of the integral in (8) and in the
following averages.
332 L. Iorio / Natural Science 2 (2010) 329-337
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 =+ 2
; (20 )
the mean anomaly changes uniformly in time at a slower
rate with respect to the unperturbed Keplerian case for
2.4. Numerical Integration of the Equations of
Motion and Explanation of the Seeming
Contradiction with the Analytical Results
At first sight, the results obtained here may be rather
confusing: if the gravitational attraction of the Sun re-
duces in time because of its mass loss the orbits of the
planets should expand (see the trajectory plotted in Fig-
ure 1, numerically integrated with MATHEMATICA),
while I obtained that the semimajor axis and the eccen-
tricity undergo secular decrements. Moreover, I found
that the Keplerian period Kep decreases, while one
would expect that the orbital period increases.
In fact, there is no contradiction, and my analytical
results do yield us realistic information on the true evo-
lution of the planetary motion. Indeed, a, e and Kep
refer to the osculating Keplerian ellipses which, at any
instant, approximate the true trajectory; it, instead, is not
an ellipse, not being bounded. Let us start at from
the osculating pericentre of the Keplerian ellipse corres-
ponding to chosen initial conditions: let us use a helio-
centric frame with the x axis oriented along the osculat-
ing pericentre. After a true revolution, i.e. when the true
Figure 1. Black continuous line: true trajectory obtained by
numerically integrating with MATHEMATICA the perturbed
equations of motion in Cartesian coordinates over 2 yr; the
disturbing acceleration (1) has been adopted. The planet starts
from the perihelion on the x axis. Just for illustrative purposes,
a mass loss rate of the order of 102 yr1 has been adopted
for the Sun; for the planet initial conditions corresponding to a
= 1 AU, e = 0.8 have been chosen. Red dashed line: unper-
turbed Keplerian ellipse at =0=. Blue dash-dotted line:
osculating Keplerian ellipse after the first perihelion passage.
As can be noted, its semimajor axis and eccentricity are clearly
smaller than those of the initial unperturbed ellipse. Note also
that after 2 yr the planet has not yet reached the perihelion as it
would have done in absence of mass loss, i.e. the true orbital
period is longer than the Keplerian one of the osculating red
radius vector of the planet has swept an angular interval
of 2, the planet finds itself again on the x axis, but at a
larger distance from the starting point because of the
orbit expansion induced by the Sun's mass loss. It is not
difficult to understand that the osculating Keplerian el-
lipse approximating the trajectory at this perihelion pas-
sage is oriented as before because there is no variation of
the (osculating) argument of pericentre, but has smaller
semimajor axis and eccentricity. And so on, revolution
after revolution, until the perturbation theory can be ap-
plied, i.e. until  
1. In Figure 1 the situ-
ation described so far is qualitatively illustrated. Just for
illustrative purposes I enhanced the overall effect by
assuming 
102 yr 1 for the Sun; the initial
conditions for the planet correspond to an unperturbed
Keplerian ellipse with a = 1 AU, e = 0.8 with the
present-day value of the Sun's mass in one of its foci. It
is apparent that the initial osculating red dashed ellipse
has larger a and e with respect to the second osculating
blue dash-dotted ellipse. Note also that the true orbital
period, intended as the time elapsed between two con-
secutive crossings of the perihelion, is larger than the
unperturbed Keplerian one of the initial red dashed os-
culating ellipse, which would amount to 1 yr for the
Earth: indeed, after 2 yr the planet has not yet reached
the perihelion for its second passage.
Now, if I compute the radial change () in the
osculating radius vector as a function of the eccentric
anomaly E I can gain useful insights concerning how
much the true path has expanded after two consecutive
perihelion passages. From the Keplerian expression of
the Sun-planet distance in (4) one gets the radial com-
ponent of the orbital perturbation expressed in terms of
the eccentric anomaly E
()=(1cos )cos 
+sin; (21)
It agrees with the results obtained in [21]. Since
sin cos
sin cos
1cos ,
=+sin 
1cos =1
()=2+2(cos  1)
()=12sin (sincos )
It follows ()=
[()+()], (24)
L. Iorio / Natural Science 2 (2010) 329-337 333
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2(sin cos )+
+sin 2+2(cos 1)
12sin 2(sin cos )
(1cos )2
, (25)
sin cos
cos sin
It turns out from (25) and (26) that, for E > 0, (E)
never vanishes; after one orbital revolution, i.e. after that
an angular interval of 2 has been swept by the (osclat-
ing) radius vector, a net increase of the radial (osculating)
distance occurs according to5
(1) (27)
This analytical result is qualitatively confirmed by the
difference6 () between the radial distances obtained
from the solutions of two numerical integrations of the
equations of motion over 3 yr with and without  
; the
initial conditions are the same. For illustrative purposes I
used a = 1 AU, e = 0.01,  
=0.1 yr1. The result
is depicted in Figure 2.
Note also that (25) and (26) tell us that the shift at the
aphelion is ()=1
1(2), (28)
in agreement with Figure 1 where it is 4.5 times larger
than the shift at the perihelion.
Since Figure 1 tells us that the orbital period gets
larger than the Keplerian one, it means that the true orbit
must somehow remain behind with respect to the Keple
rian one. Thus, a negative perturbation  in the trans-
verse direction must occur as well; see Figure 3.
Let us now analytically compute it. According to [21],
it can be used
12+12+(+cos ). (29)
By recalling that, in the present case, = 0 and
1cos, (30)
it is possible to obtain from (22) and (30)
Figure 2. Difference () between the radial distances ob-
tained from the solutions of two numerical integrations with
MATHEMATICA of the equations of motion over 3 yr with
and without 
; the initial conditions are the same. Just for
illustrative purposes a mass loss rate of the order of  
0.1 yr1 has been adopted for the Sun; for the planet initial
conditions corresponding to a = 1 AU, e = 0.01 have been
chosen. The cumulative increase of the Sun-planet distance
induced by the mass loss is apparent.
Figure 3. Radial and transverse perturbations  and  of
the Keplerian radius vector (in blue); the presence of the
transverse perturbation  makes the real orbit (in red) lag-
ging behind the Keplerian one.
()=sin (cossin ),
(1cos )
[(1 + )(cos 1)+sin ],
+ 2(cos 1),
()=sin 
(1+)(1cos )sin
It turns out from (31) and (32) that, for > 0,
() never vanishes; at the time of perihelion passage
1 <0. (33)
According to (25) and (26),
r(0) = 0.
Strictly speaking,
r and the quantity plotted in Figure 2
are different
objects, but, as the following discussion will clarify, I can assume that,
in practice, they are the same.
334 L. Iorio / Natural Science 2 (2010) 329-337
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This means that when the Keplerian path has reached
the perihelion, the perturbed orbit is still behind it. Such
features are qualitatively confirmed by Figure 1. From a
vectorial point of view, the radial and transverse pertur-
bations to the Keplerian radius vector r yield a correc-
tion =
, (34 )
so that pert =+. (35)
The length of is
()=()2+()2. (36)
Eqs.27 and 31 tell us that at perihelion it amounts to
(2)=(2)1 + 42(1+)
(1)3 . (37)
The angle ξ between and r is given by
tan () = ()
(); (38)
at perihelion it is
tan (2)= 21+
, (39)
i.e. ξ is close to 90 deg; for the Earth it is 81.1 deg. Thus,
the difference between the lengths of the perturbed
radius vector pert and the Keplerian one r at a given
instant amounts to about
cos ; (40)
in fact, this is precisely the quantity determined over 3 yr
by the numerical integration of Figure 2. At the perihe-
lion I have
=(2)1 + 42(1 + )
(1)3 cos . (41)
Since for the Earth
1 + 42(1+)
(1)3 cos =1.0037, (42)
it holds (2). (43)
This explains why Figure 2 gives us just .
Concerning the observationally determined increase
of the Astronomical Unit, more recent estimates from
processing of huge planetary data sets by Pitjeva point
towards a rate of the order of 102 m yr 1 [22,23]. It
may be noted that my result for the secular variation of
the terrestrial radial position on the line of the apsides
would agree with such a figure by either assuming a
mass loss by the Sun of just 9 × 1014 yr1 or a
decrease of the Newtonian gravitational constant.
. Such a value for the temporal varia-
tion of G is in agreement with recent upper limits
from Lunar Laser Ranging [24].
. This possibility is envisaged in [25] whose
authors use  
by speaking about a small
radial drift of (6 ± 13)×102 m yr1 in an orbit at 1
In this Section 1 will not consider other effects which
may affect the final evolution of the Sun-Earth system
like the tidal interaction between the Earth and the tidal
bulges of the giant solar photosphere, and the drag fric-
tion in the motion through the low chromosphere [1].
For the Earth, by assuming the values a = 1.00000011
AU, e = 0.01671022 at the epoch J2000 (JD 2451545.0)
with respect to the mean ecliptic and equinox of J2000
and  
=9 × 1014 yr 1, (24) yields
(2)=1.3 × 102m. (44)
This means that at every revolution the position of the
Earth is shifted along the true line of the apsides (which
coincides with the osculating one because of the absence
of perihelion precession) by 1.3 cm. This result is con-
firmed by our numerical integrations and the discussion
of Section 2; indeed, it can be directly inferred from
Figure 2 by multiplying the value of  at t = 1 yr by
9 × 1013. By assuming that the Sun will continue to
lose mass at the same rate for other 7.58 Gyr, when it
will reach the tip of the RGB in the HR diagram [1], the
Earth will be only 6.7 × 104 AU more distant than
now from the Sun at the perihelion. Note that the value
9 × 1014 yr 1 is an upper bound on the magnitude of
the Suns mass loss rate; it might be also smaller [1] like,
e.g., 7 × 1014 yr1 which would yield an increment
of 5.5 × 104 AU. Concerning the effect of the other
planets during such a long-lasting phase, a detailed cal-
culation of their impact is beyond the scope of the
present paper. By the way, I wish to note that the depen-
dence of (2) on the eccentricity is rather weak;
indeed, it turns out that, according to (24), the shift of
the perihelion position after one orbit varies in the range
1.1 1.3 cm for 00.1 . Should the interaction
with the other planets increase notably the eccentricity,
the expansion of the orbit would be even smaller; indeed,
for higher values of e like, e.g., e = 0.8 it reduces to
about 3 mm. By the way, it seems that the eccentricity of
the Earth can get as large as just 0.02 0.1 [26-28] over
timescales of 5 Gyr due to the N-body interactions
with the other planets. In Table 1, I quote the expansion
of the orbits of the other planets of the solar system as
L. Iorio / Natural Science 2 (2010) 329-337 335
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It is interesting to note that Mercury7 and likely Venus
are fated at the beginning of the RGB; indeed, from
Figure 2 of [1] it turns out that the Suns photosphere
will reach about 0.5-0.6 AU, while the first two planets
of the solar system will basically remain at 0.38 AU and
0.72 AU, respectively, being the expansion of their orbits
negligible according to Table 1.
After entering the RG phase things will dramatically
change because in only 1 Myr the Sun will reach the
tip of the RGB phase loosing mass at a rate of about
2 × 107 yr1 and expanding up to 1.20 AU [1]. In
the meantime, according to our perturbative calculations,
the perihelion distance of the Earth will increase by 0.25
AU. I have used as initial conditions for , a and e their
final values of the preceding phase 7.58 Gyr-long. In
Table 2, I quote the expansion experienced by the other
planets as well; it is interesting to note that the outer
planets of the solar system will undergo a considerable
increase in the size of their orbits, up to 7.5 AU for
Neptune, contrary to the conclusions of the numerical
computations in [29] who included the mass loss as well.
I have used as initial conditions the final ones of the pre-
vious MS phase. Such an assumption seems reasonable
for the giant planets since their eccentricities should be
left substantially unchanged by the mutual N-body inte-
ractions during the next 5 Gyr and more [26-28]; con-
cerning the Earth, should its eccentricity become as
Table 1. Expansion of the orbits, in AU, of the eight planets of
the solar system in the next 7.58 Gyr for
=9 ×
1014 yr1. I have neglected mutual N-body interactions.
2 × 104
5 × 104
7 × 104
9 × 104
3 × 103
6 × 103
1 × 10
2 × 102
Table 2. Expansion of the orbits, in AU, of the eight planets of
the solar system in the first 1 Myr of the RGB for
2 × 107 yr1. I have neglected mutual N-body interactions
and other phenomena like the effects of tidal bulges and chro-
mospheric drag for the inner planets.
7 × 102
1.8 × 101
2.5 × 101
3.4 × 101
large as 0.1 due to the N-body perturbations [26-28],
after about 1 Myr its radial shift would be smaller
amounting to 0.22 AU. Mutual N-body interactions have
not been considered. Thus, orbital hardly preventing our
planet to escape from engulfment in the expanding solar
photosphere. Concerning the result for the Earth, it must
be pointed out that it remains substantially unchanged if
I repeat the calculation by assuming a circularized orbit
during the entire RGB phase. Indeed, it is possible to
show that by adopting as initial values of a and the
final ones of the previous phase I get that after 1.5
Myr  has changed by 0.30 AU. Note that my results
are in contrast with those in [1] whose authors obtain
more comfortable values for the expansion of the Earths
orbit, assumed circular and not influenced by tidal and
frictional effects, ranging from 1.37 AU (| 
|=7 ×
1014 yr 1) to 1.50 AU (| 
|=8 × 1014 yr 1) and
1.63 AU (| 
|=9 × 1014 yr 1). However, it must be
noted that such a conclusion relies upon a perturbative
treatment of (1) and by assuming that the mass loss rate
is constant throughout the RGB until its tip; in fact, dur-
ing such a Myr the term (
)  would get as large as
2 × 101. In fact, by inspecting Figure 4 of [1] it ap-
pears that in the last Myr of the RGB a moderate varia-
tion of
occurs giving rise to an acceleration of
the order of
1013 yr2. Thus, a further qua-
dratic term of the form
2 (45)
should be accounted for in the expansion of (1). A per-
turbative treatment yields adequate results for such a
phase 1 Myr long since over this time span (45) would
amount to 5 × 102. However, there is no need for
detailed calculations: indeed, it can be easily noted that
the radial shift after one revolution is
. (46)
After about 1 Myr (46) yields a variation of the order
of 109 AU, which is clearly negligible.
I started in the framework of the two-body Newtonian
dynamics by using a radial perturbing acceleration linear
in time and straightforwardly treated it with the standard
Gaussian scheme. I found that the osculating semima-
jor axis a, the eccentricity e and the mean anomaly
secularly decrease while the argument of pericentre ω
remains unchanged; also the longitude of the ascending
node Ω and the inclination I are not affected. The radial
distance from the central body, taken on the fixed line of
the apsides, experiences a secular increase . For the
Earth, such an effect amounts to about 1.3 cm yr 1. By
numerically integrating the equations of motion in Car-
7It might also escape from the solar system or collide with Venus over
3.5 Gyr from now
336 L. Iorio / Natural Science 2 (2010) 329-337
Copyright © 2010 SciRes. OPEN ACCESS
tesian coordinates I found that the real orbital path ex-
pands after every revolution, the line of the apsides does
not change and the apsidal period is larger than the un-
perturbed Keplerian one. I have also clarified that such
results are not in contrast with those analytically ob-
tained for the Keplerian orbital elements which, indeed,
refer to the osculating ellipses approximating the true
trajectory at each instant. I applied our results to the
evolution of the Sun-Earth system in the distant future
with particular care to the phase in which the Sun,
moved to the RGB of the HR, will expand up to 1.20 AU
in order to see if the Earth will avoid to be engulfed by
the expanded solar photosphere. My answer is negative
because, even considering a small acceleration in the
process of the solar mass-loss, it turns out that at the end
of such a dramatic phase lasting about 1 Myr the perihe-
lion distance will have increased by only  0.22-
0.25 AU, contrary to the estimates in [1] whose authors
argue an increment of about 0.37-0.63 AU. In the case of
a circular orbit, the osculating semimajor axis remains
unchanged, as confirmed by a numerical integration of
the equations of motion which also shows that the true
orbital period increases and is larger than the unper-
turbed Keplerian one which remains fixed. Concerning
the other planets, while Mercury will be completely en-
gulfed already at the end of the MS, Venus might survive;
however, it should not escape from its fate in the initial
phase of the RGB in which the outer planets will expe-
rience increases in the size of their orbits of the order of
1.2- 7.5 AU.
As a suggestion to other researchers, it would be very
important to complement my analytical two-body calcu-
lation by performing simultaneous long-term numerical
integrations of the equations of motion of all the major
bodies of the solar system by including a mass-loss term
in the dynamical force models as well to see if the
N-body interactions in presence of such an effect may
substantially change the picture outlined here. It would
be important especially in the RGB phase in which the
inner regions of the solar system should dramatically
I thank Prof. K.V. Kholshevnikov, St. Petersburg State University for
useful comments and references.
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