Astrophys Space Sci (2016) 361:122 DOI 10.1007/s10509-016-2684-5
O R I G I N A L A RT I C L E
The effects of solar Reimers η on the final destinies of Venus, the Earth, and Mars Jianpo Guo1,2 · Ling Lin1 · Chunyan Bai1 · Jinzhong Liu3
Received: 11 August 2015 / Accepted: 3 February 2016 © Springer Science+Business Media Dordrecht 2016
Abstract Our Sun will lose sizable mass and expand enormously when it evolves to the red giant branch phase and the asymptotic giant branch phase. The loss of solar mass will push a planet outward. On the contrary, solar expansion will enhance tidal effects, and tidal force will drive a planet inward. Will our Sun finally engulf Venus, the Earth, and Mars? In the literature, one can find a large number of studies with different points of view. A key factor is that we do not know how much mass the Sun will lose at the late stages. The Reimers η can describe the efficiency of stellar mass-loss and greatly affect solar mass and solar radius at the late stages. In this work, we study how the final destinies of Venus, the Earth, and Mars can be depending on Reimers η chosen. In our calculation, the Reimers η varies from 0.00 to 0.75, with the minimum interval 0.0025. Our results show that Venus will be engulfed by the Sun and Mars will most probably survive finally. The fate of the Earth is uncertain. The Earth will finally be engulfed by the Sun while η < 0.4600, and it will finally survive while η ≥ 0.4600. New observations indicate that the average Reimers η for solar-like stars is 0.477. This implies that Earth may survive finally. Keywords Stars: radius · Stars: mass-loss · Planets: orbit
B J. Guo
[email protected]
1
Department of Science and Technology, Puer University, Puer 665000, Yunnan Province, China
2
Open Key Laboratory of Mechanics in Yunnan Province, Puer 665000, Yunnan Province, China
3
Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011, Xinjiang Province, China
1 Introduction According to the theory of stellar structure and evolution, our Sun will expand enormously twice at the late stages of its life. The first expansion will occur when it evolves to the red giant branch (RGB) phase, and the second expansion will happen when it evolves to the asymptotic giant branch (AGB) phase. Our Sun may expand to about or even more than 200 times the present dimensions in the future. The orbital semi-major axes of Mercury, Venus, the Earth, Mars, and Jupiter are 83 R , 156 R , 215 R , 328 R , and 1 119 R , respectively. It is certain that Mercury will be overwhelmed by the Sun and Jupiter will survive finally. However, will our Sun engulf Venus, the Earth, and Mars, when it reaches the RGB-tip or the AGB-tip? The final destinies of Venus, the Earth, and Mars cannot be estimated by simply comparing their present semi-major axes and the maximum radius of the Sun. With the Sun evolving, it will lose mass and push a planet outward. Sackmann et al. (1993) first studied this issue and pointed out that the final destinies of Venus and Earth strongly depend on the efficiency of the solar mass-loss. Using Reimers’ η = 0.6 normalized from inferred mass-loss in globular cluster stars according to observations at that time, both Venus and Earth will finally survive. Silvotti et al. (2007) also thought that the Earth will finally survive. On the other hand, tidal effects will become stronger and stronger, with the Sun expanding. The tidal force will drive Venus, the Earth, and Mars inward, when the Sun evolves to the late stages. Taking the tidal interaction into account, Venus will most probably be destroyed, the Earth’s fate is controversial, Mars will most probably survive (e.g., Rasio et al. 1996; Rybicki and Denis 2001; Schröder and Smith 2008). The destinies of Venus, the Earth, and Mars are closely related to the solar mass and solar radius at the RGB-tip and
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the AGB-tip. Both the mass and the radius are bound up with the efficiency of the solar mass-loss. Mass-loss is a significant component for most stellar evolution codes, which can be parameterized by stellar physical quantities, such as the luminosity, radius, and mass. Mass-loss is often modeled to have the following Reimers functional form (Reimers 1975): dM L R M −1 = −4η × 10−13 dt L R M M yr−1 ,
2.1 The effects of solar mass-loss on planetary orbit The Sun will lose a sizable mass, which can push planets outward (e.g., Sackmann et al. 1993). With the orbital eccentricity remaining fixed, the semi-major axis a of a planetary orbit can be obtained by a=
(1)
where η is called Reimers’ parameter, which describes the efficiency of the mass-loss. Veras and Wyatt (2012) studied the solar system at postmain-sequence, with the Reimers η varying from 0.2 to 0.8. Their results show that Reimers η can greatly affect the solar system’s post-main-sequence stability boundary. Through measuring the efficiency of the RGB mass-loss for solarlike stars in 56 well-studied Galactic globular clusters, it has been found that the values of the Reimers η for these stars vary from 0.281 to 0.749 (McDonald and Zijlstra 2015). The average value is 0.477 and the standard deviation is 0.070. Mass-loss mechanisms on the RGB have very little metallicity dependence. Schröder and Cuntz (2005) updated the traditional Reimers law using two extra terms: dM L R M −1 = −4ηSC × 10−13 dt L R M 3.5 Teff g × 1+ M yr−1 , 4000 K 4300g
2 Several effects on planetary orbit
(2)
where ηSC is the mass-loss parameter of Schröder and Cuntz (2005), g and Teff are the star’s surface gravity and effective temperature, respectively, and g is the current value of the solar surface gravity. There are two globular clusters, named NGC 5904 and NGC 5927, their metallicities are 0.001 and 0.01, respectively (Piotto et al. 2002). Equation (2) can reproduce the two globular clusters with the same ηSC . In this work, we only calculate the solar evolution. Hence, we adopt the mass-loss mode of Reimers’ law, and study how the final destinies of Venus, the Earth, and Mars might be depending on the Reimers η chosen. The outline of this paper is as follows: we show several effects on planetary orbit in Sect. 2, describe our methods for stellar evolution code in Sect. 3, show our results in Sect. 4, give several discussions in Sect. 5, and then finally in Sect. 6 we give a conclusion.
M a0 , M
(3)
where M denotes the mass of the Sun with evolution and a0 denotes the semi-major axis of the planetary orbit at present. Equation (3) is widely adopted (e.g., Sackmann et al. 1993; Silvotti et al. 2007), and it holds while the solar system planets are within about 500 AU (Veras et al. 2011). 2.2 The tidal effects on planetary orbit The Sun will expand about 200 times in the RGB phase and this makes its rotation period extend to thousands of years or even longer (e.g., Schröder and Smith 2008; Guo et al. 2016). The orbital periods of Venus, the Earth, and Mars are 0.61, 1.00, and 1.86 years, respectively. Comparatively speaking, the Sun in the RGB phase can be considered as no spin, and the tidal force will drive planets inward (e.g., Rybicki and Denis 2001; Kunitomo et al. 2011). Considering the effects of tide and mass-loss on a planetary orbit, Kunitomo et al. (2011) integrated the variable equation of planetary orbit: 8 M˙ Mp 1 da R k Mp = −6 F e2 − , (4) 1+ a dt T M M a M where Mp is planet’s mass, M and R are mass and radius of the Sun with evolution, respectively. F (e2 ) is the ellipticity modification to the tidal effects (e.g., Hut 1981; Kunitomo et al. 2011), which is expressed by 1+ F e2 =
31 2 2 e
185 6 4 + 255 8 e + 16 e + (1 − e2 )15/2
25 8 64 e
.
(5)
k is the apsidal motion constant and T is the eddy turnover timescale. Following Rasio et al. (1996) and Villaver and Livio (2009), Kunitomo et al. (2011) adopted the turbulent viscosity and gave the expression of k, k=
f Menv . 6 M
(6)
Here, Menv is the mass in the convective envelope and f can be expressed by 2 P f = min 1, . (7) 2T
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Here the planetary orbital period P can be expressed by
a3 , (8) P = 2π G(M + Mp ) and the eddy turnover timescale T may be written
Menv (R − Renv )2 T= 3L
1 3
.
(9)
Renv is the radius at the base of the convective zone and L is the luminosity. According to the theory of stellar structure and evolution, the whole envelope is convective when the Sun evolves to the RGB phase. Eggleton’s code (see below) can gives the same result. Therefore, Menv and Renv are adopted as envelope mass and core radius of the Sun with evolution in our calculation, respectively.
(10)
where Rp is the planet’s radius, M is the solar mass-loss, and mwd is the wind mass colliding with a planet. The planet can get the maximum radial velocity vr , provided that mwd blows to a planet with vwd and bounces back with −vwd . According to the law of momentum conservation, 2mwd vwd = Mp vr .
(11)
Combining Eq. (10) with (11), we obtain vr =
M Rp2 2Mp a 2
vwd .
(12)
In our calculation, the core mass of the Sun at the RGB-tip is about 0.47 M . Therefore, the maximum mass-loss of the Sun in its whole lifetime is about 0.53 M . The velocity of the solar wind is variable with solar evolution, and the maximum velocity is about 700 km/s (Suzuki 2013). As a planet moves around Sun, the relation between mechanical energy and semi-major axis is E=−
GM Mp . 2a
Semi-major axis (AU)
Mass (M⊕ )
Radius (R⊕ )
a/a
0.3873
0.0554
0.383
1.714 × 10−3
0.7239
0.8150
0.949
4.572 × 10−5
1.0009
1.0000
1.000
1.416 × 10−5
1.5250
0.1075
0.532
2.776 × 10−5
5.2075
317.90
11.20
1.566 × 10−8
9.5474
95.180
9.410
1.412 × 10−8
19.1991
14.630
4.060
2.548 × 10−9
30.0848
17.240
3.880
3.977 × 10−10
conservation of mechanical energy, one can derive that GM Mp 1 GM Mp =− + Mp vr2 . 2(a + a) 2a 2
(14)
Solving Eq. (14), we obtain
Provided that the solar wind is isotropic, the relationship between M and mwd can be expressed by M mwd = , 4πa 2 πRp 2
Table 1 The values of a/a for eight planets’ orbit, which are caused by the effects of solar wind impulse. The values of the semi-major axes for eight planets are cited from Allen (1973)
−
2.3 The impulsive effects of solar wind on planetary orbit
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(13)
The planet’s radial velocity increases by vr , and the semimajor axis a will change to a + a. According to the law of
a avr2 = . a GM
(15)
Putting the relevant data into Eq. (15), we can get the a/a for eight planets of solar system, given in Table 1. All the values of a/a are so small that the impulsive effects of the solar wind on the planetary orbit can be negligible. 2.4 Other effects on planetary orbit The Sun will lose a great deal of mass in the RGB phase and AGB phase, some wind mass will collide with planets or even accrete onto planets. Duncan and Lissauer (1998) implied that each planet probably will collide with significantly less than 1% of its mass over the Sun’s entire RGB phase. Spiegel and Madhusudhan (2012) pointed out that the stars’ wind mass will tend to accrete onto Jovian-mass planets. Even for Jupiter, the accreted mass is only about 10−5 the Jovian mass. Although the accreted mass might be large compared with a planet’s atmosphere mass, it is a small fraction of the planet’s total mass and has a negligible effects on the orbit (Villaver and Livio 2009). Whether the stellar mass-loss is isotropic will affect the accreted mass for a planet (Spiegel and Madhusudhan 2012). However, outflows are often modeled to be isotropic. The anisotropic stellar mass-loss is realistic for fast rotators, bipolar jets, and supernovae (Veras et al. 2013). During the RGB phase and AGB phase, our Sun will rotate extremely slowly (e.g., Schröder and Smith 2008; Guo et al. 2016). Therefore, solar mass-loss may be deemed to be isotropic.
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3 Methods for stellar evolution code 3.1 Input physics of stellar evolution We use the stellar evolution code by Eggleton (1971, 1972, 1973), which has been updated with input physics over the last four decades (Han et al. 1994; Pols et al. 1995, 1998). The convective overshooting parameter has been set at δOV = 0.12 (Pols et al. 1997; Schröder et al. 1997). The standard solar metal mixture is used by Grevesse and Sauval (1998) and the solar metallicity has been assumed to be 0.02. The initial hydrogen mass fraction is 0.70 (Pols et al. 1998; Jiang et al. 2010). We use OPAL high temperatures opacity tables (Iglesias and Rogers 1996; Eldridge and Tout 2004) in the range of 4.00 < log(T/K) ≤ 8.70, and the new Wichita state low temperature molecular opacity tables (Ferguson et al. 2005) in the range of 3.00 ≤ log(T/K) ≤ 4.00. We have made the opacity tables match well with Eggleton’s code (Chen and Tout 2007; Guo et al. 2008). A Reimers type mass-loss has been adopted in our calculation. Instead of η, 4η serves as an input parameter in Eggleton’s code. The value of 4η can range from 0.00 to 4.00, with the minimum interval 0.01. This means that η can range from 0.00 to 1.00, with the minimum interval 0.0025. η = 0.00 means no mass-loss, and the maximum observed Reimers η of solar-like stars in the RGB phase is 0.749 (McDonald and Zijlstra 2015). Therefore, the value of the Reimers η ranges from 0.00 to 0.75 in our calculation. 3.2 Constructing the evolutionary mode of the Sun at ZAHB Using Eggleton’s code, we can easily simulate the solar evolution from the zero age main sequence to the RGB-tip. However, the core for the Sun in the RGB phase is degenerate; it cannot pass through the helium flare. Therefore, we have to construct the evolutionary mode of the Sun at the zero age horizontal branch (ZAHB), using an intermediate mass star. Our methods for constructing the ZAHB models are similar to Zhang et al. (2009) and Guo et al. (2010). The concrete steps are as follows. First of all, we make the Sun with a designated Reimers η normally evolve to the RGB-terminal and record the total mass, core mass, and element abundance profile of the Sun at the RGB-terminal. Second, we make an intermediate mass star, whose mass is 3.0 M , automatically evolve to the RGB-bottom. At the RGB-bottom, hydrogen on the core will be exhausted and helium burning on the core does not start yet. Moreover, the core mass at the RGB-bottom is generally very small, about 0.3 M . Third, we shut down helium burning, and the helium core will grow gradually. Fourth, we also shut down hydrogen burning and let the Sun
Fig. 1 Solar mass and solar radius at the RGB-tip, with different Reimers η. The asterisks denote masses, in units of M . The points denote radii, in units of AU, 1 AU = 215 R
rapidly lose mass, once the core mass reaches the recorded value as in the first step. The star will sharply lose envelope mass and the core mass is invariable. Fifth, we shut down the rapid mass-loss, once the total mass reaches the recorded value as in the first step. Sixth, we want to totally replace the element abundance profile by that profile recorded in the first step. Unfortunately, this will make Eggleton’s code stop. Therefore, we have to take the second best, and only replace the element abundance profile in the convective region. Seventh, we open hydrogen burning and helium burning to get a ZAHB model, with the same total mass, core mass, and element abundance profile in the convective region as the RGBterminal. Then it enters into the horizontal branch (HB) phase. Changing the Reimers η and repeating the above steps, one can get solar evolutionary data at the HB phase, with different Reimers η.
4 Results 4.1 The destinies of Venus, the Earth, and Mars during solar RGB phase The Sun will lose a nice bit of its mass in the RGB phase, its radius will expand to dimensions about 200 times larger than the current state. Both solar mass and solar radius at the RGB-tip are different, changing with the Reimers η; see Fig. 1. The solar surface will achieve the present orbits of Venus, the Earth, and Mars, when the Sun evolves to RGB phase. The solar mass-loss will push all the three planets outward. Will Venus, the Earth or Mars be engulfed by the Sun in the RGB phase? It depends on whether the Sun’s radius is larger than the planetary perihelion distance.
Final destinies of Venus, the Earth, Mars
Fig. 2 The perihelion distances r1 of Venus, the Earth, and Mars, when the Sun evolves to late RGB phase, with Reimers’ η = 0.4125. The full line denotes Sun’s radius. The dot–dash line, the dashed line, and the dotted line denote the perihelion distances of Venus, Earth, and Mars, respectively
Fig. 3 Solar mass and solar radius at the AGB-tip, with different Reimers η. The asterisks denote masses, in units of M . The points denote radii, in units of AU
The orbital ellipticities of Venus, the Earth, and Mars are 0.007, 0.017, and 0.093, respectively. The relation between perihelion distance r1 and semi-major axis a is r1 = a(1 − e). According to our estimate, Venus will certainly be engulfed and Mars will not be engulfed by the Sun in the RGB phase, with the Reimers η varying from 0.00 to 0.75. The destiny of the Earth is uncertain. Earth will not be engulfed by the Sun in the RGB phase while η ≥ 0.4125; see Fig. 2. 4.2 The destinies of the Earth and Mars during solar AGB phase During the AGB phase, the Sun will also expand vigorously. Therefore, it does not mean that the planet will also not be
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Fig. 4 The perihelion distances of Venus, the Earth, and Mars, when the Sun evolves to late RGB phase and AGB phase, with Reimers’ η = 0.1850. The full line, the dot–dash line, the dashed line, and the dotted line have the same meaning as in Fig. 2
engulfed by the Sun during the AGB phase, if a planet is not be engulfed by the Sun during RGB phase. We construct the ZAHB model and calculate the evolution of the Sun after the helium flare. Furthermore, we give the solar mass and solar radius at the AGB-tip, with the Reimers η varying from 0.00 to 0.70; see Fig. 3. The ZAHB model for the Sun cannot easily be constructed, when η > 0.70. The Sun will lose a great deal of mass in the RGB phase and its envelope will become very thin. For getting the evolutionary data with the Reimers η larger than 0.70, we should revise the stellar evolution code. Thus may cause systematic errors. Therefore, we do not give the evolutionary data for the Sun during the HB and AGB phase, with η > 0.70. Once η < 0.1850, Mars will be engulfed during the AGB phase. And Mars will just survive, when η = 0.1850; see Fig. 4. While η = 0.4125, the Earth will just not be engulfed by the Sun during the RGB phase, but it will be engulfed by the Sun during the AGB phase; see Fig. 5. Therefore, the Earth will be engulfed by the Sun in the RGB phase and Mars will finally survive while 0.1850 ≤ η < 0.4125. While η = 0.4600, the Earth will just right not be engulfed by the Sun during the AGB phase; see Fig. 6. Hence, the Earth will be engulfed by the Sun during the AGB phase and Mars will finally survive while 0.4125 ≤ η < 0.4600. Both the Earth and Mars will finally survive while η ≥ 0.4600. McDonald and Zijlstra (2015) measured the efficiency of the RGB mass-loss for solar-like stars in 56 well-studied Galactic globular clusters. It has been found that the values of the Reimers η for these stars vary from 0.281 to 0.749.
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Fig. 5 The perihelion distances of Venus, the Earth, and Mars, when the Sun evolves to the late RGB phase and the AGB phase, with Reimers’ η = 0.4125. The full line, the dot–dash line, the dashed line, and the dotted line have the same meaning as in Fig. 2
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Fig. 7 The perihelion distances of Venus, the Earth, and Mars, when the Sun evolves to the late RGB phase and the AGB phase, with Reimers’ η = 0.4775. The full line, the dot–dash line, the dashed line, and the dotted line have the same meaning as in Fig. 2
5 Discussions 5.1 Comparison of our evolutionary data with previous work
Fig. 6 The perihelion distances of Venus, the Earth, and Mars, when the Sun evolves to the late RGB phase and AGB phase, with Reimers’ η = 0.4600. The full line, the dot–dash line, the dashed line, and the dotted line have the same meaning as in Fig. 2
According to our estimate, Mars will finally survive while η ≥ 0.1850. Therefore, we can deduce that Mars will not be engulfed by the Sun finally. According to our calculation, the Earth will finally be destroyed while η < 0.4600 and survive while η ≥ 0.4600. Hence, the final fate of the Earth is indeterminate. The average Reimers η of solar-like stars is 0.477. Therefore, we should calculate the evolution of the Sun using η = 0.477. As the minimum interval of Reimers’s η in Eggleton’s code is 0.0025, the proximate value to 0.477 is 0.4775. While η = 0.4775, the Earth will finally survive; a more intuitive view can be seen in Fig. 7.
In our calculation, the mass of the Sun at the RGB-tip is 0.7428 M and the radius is 196.03 R while η = 0.4775. At the AGB-tip, the mass of the Sun is 0.6190 M and the radius is 195.86 R . Using the YNEV code and the method of Zhang (2015), the mass of the Sun at the RGB-tip is 0.60 M and the radius is 220 R while η = 0.477. Rasio et al. (1996) pointed out that the radius of the Sun at the RGB-tip is 195 R while η = 0.60. On the basis of our calculation, the value is 200.55 R while η = 0.60. The Sun will expand enormously during the RGB phase and the AGB phase. Generally speaking, the Sun’s radius will attain its maximum at the AGB-tip (e.g., Sackmann et al. 1993; Rybicki and Denis 2001). However, the Sun at the AGB-tip will not reach its size at the RGB-tip, according to the calculation of Schröder and Smith (2008). In their opinion, the main reason is that a more significant amount of mass has already lost in the RGB phase. Actually, both of the two viewpoints are right, even though they seem to be contradictory to each other. The crucial factor is how much mass will be lost in the RGB phase. The Sun will reach its maximum size at the AGB-tip, provided that it loses less mass in the RGB phase. On the contrary, the Sun will reach its maximum size at the RGB-tip, if it loses significant wind mass in the RGB phase. The mass-loss can also be quantitatively described by Reimers η. The turnoff point for Reimers η is located between 0.4750 and 0.4775, according to our calculation. The Sun will reach its maximum size at the AGB-tip, if η ≤ 0.4750; see Figs. 4, 5, and 6. On the contrary, the Sun
Final destinies of Venus, the Earth, Mars
will reach its maximum size at the RGB-tip, if η ≥ 0.4775; see Fig. 7. 5.2 The mass-loss rate of the Sun during the AGB phase In our calculation, the Reimers η of the Sun during the AGB phase is the same as during RGB phase. Unfortunately, this assumption may be inconsistent with observations. Renzini (1981) showed that Reimers’ type mass-loss cannot make a star generate superwind during the AGB phase. During the AGB phase, the Reimers relation gives massloss rates that increase too slowly, take too long, and reach values too small (e.g., Willson 2000; Bowen and Willson 1991). Furthermore, Reimers’ mass-loss rates do not allow for the formation of planetary nebulae (e.g., Renzini 1981; Iben and Renzini 1983; Schoenberner 1983), and they fail to reproduce the high mass-loss rates observed in AGB stars (e.g., Winters et al. 2000; Wood 1979) and the increase in mass-loss during the AGB ascent (e.g., Wood 1979; Willson 2000). Therefore, Mustill and Villaver (2012) definitely pointed out that the commonly used Reimers η should not apply to the AGB phase. From Fig. 3, it can be seen that both the solar mass and the solar radius at the AGB-tip decrease with the Reimers η increasing, once η ≥ 0.15. The last paragraph shows that the Sun will lose more wind mass than would agree with Reimers’ mass-loss rates during the AGB phase. More massloss makes both the solar mass and the solar radius at the AGB-tip smaller. Furthermore, more mass-loss will push a planet outward more effectively. A smaller radius leads to a weaker tidal effect. Therefore, the Sun loses more mass during the AGB phase and this means that a planet becomes safer. Venus will be engulfed by the Sun during the RGB phase. Thus, that the Sun loses more mass during the AGB phase does not affect the final destiny of Venus. It is almost certain that Mars will finally survive, and more mass-loss by the Sun during the AGB phase will push Mars more outward. While η < 0.4125, the Earth will be engulfed by the Sun in the RGB phase, so that it becomes meaningless that the Sun loses more mass during the AGB phase. While η ≥ 0.4600, the Earth will finally survive, and more mass-loss by the Sun during the AGB phase will make Earth safer. According to our calculation, the Earth will be engulfed by the Sun during the AGB phase while 0.4125 ≤ η < 0.4600. In fact, Sun will lose more mass during the AGB phase, which pushes the Earth more outward and the Earth may escape the destiny of being engulfed by the Sun. Therefore, the Earth may finally survive while 0.4125 ≤ η < 0.4600. In other words, the probability that Earth will finally survive becomes larger. 5.3 The ellipticity evolution of planetary orbit Tidal effects not only drive a planet inward, but also make its orbit become circular, when the host star evolves to the RGB
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phase or the AGB phase. In other words, tidal effects are able to reduce the planetary orbital eccentricity. The semi-major axis evolution being coupled with the ellipticity evolution is adopted in some literature (e.g., Mustill and Villaver 2012; Villaver et al. 2014). In our calculation, we adopt a fixed ellipticity, the main reasons being as follows. The orbital ellipticities of Venus and Earth are 0.007 and 0.017, respectively. The ellipticity is so small that the ellipticity modification to the tidal effects can be neglected; see Eq. (5). Therefore, it is accepted that the ellipticity evolution is not taken into consideration for Venus and Earth. The orbital ellipticity of Mars is 0.093 and the ellipticity modification to tidal effects cannot be neglected. However, the semi-major axis of Mars is larger and the mass of Mars is smaller, which leads to a faint tidal effect; see Eq. (4). According to our calculation, Mars will finally survive while η ≥ 0.1850, if we account for tidal effects. Mars will finally not be engulfed by the Sun while η ≥ 0.1825, if we do not account for tidal effects. Hence, it is also accepted that the ellipticity evolution is not taken into consideration for Mars.
6 Conclusion First, we study several effects on a planetary orbit, such as the effects of solar mass-loss, tidal effects, the impulsive effects of the solar wind, the effects of the mass accreted onto a planet and the effects of the anisotropy. The impulsive effects, the mass accretion effects, and the anisotropic effects can be negligible. Second, we describe the input physics of Eggleton’s code and introduce the construction of the ZAHB mode. Third, we study the final destinies of Venus, the Earth, and Mars, changing with the solar Reimers η. It is found that Venus will certainly be engulfed by the Sun in the RGB phase and Mars will finally survive. The final fate of the Earth is indeterminate. The Earth will finally survive while η ≥ 0.4600. Fourth, we compare our results with previous work, and we discuss the mass-loss rate of the Sun during the AGB phase and the ellipticity evolution of planetary orbit. More data about this work is available upon request to
[email protected] or
[email protected]. Acknowledgements This work is partially supported by the Scientific Research Foundation of Yunnan Educational Department (Grants No. 2012Y and 2014Y), CAS “Light of West China” Program (2015XBQN-A-02), the Natural Science Foundation of Puer University (Grant No. K2015030) and the National Natural Science Foundation of China (Grant No. 11265012). We would like to thank an anonymous referee for a careful review and pertinent comments. Jianpo Guo would also like to thank tutors and classmates in Yunnan Observatory for help.
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