Figure 2. Effects of Earth and Sun gravity on the circular lunar polar orbit.

tional model with degree and order 50 is used. The lunar gravity field, LP150Q, is a 150th spherical harmonic degree and order model. This gravity field was developed using the available data from past US missions to Moon including the Lunar Orbiter missions, the Apollo 15 and Apollo 16 sub-satellites, Clementine, and the Lunar Prospector Doppler and range data [14].

In the present study, the radius of moon is taken to be 1738 km. The values of and (gravitational constants of Moon, Earth and Sun, respectively) used in the simulations are 4902.800076 km^{3}/s^{2}, 398,600.436233 km^{3}/s^{2} and 132,712,440,040.944 km^{3}/s^{2}, respectively. For every simulation, the orbit is propagated taking January 1, 2010 (TDB) as initial epoch. Lifetime is calculated as the time from the initial epoch to the time at which the periapsis altitude becomes zero.

3.2. Lifetime of Low Altitude Lunar Orbits

The estimation of lifetime of a lunar orbit strongly depends on the lunar gravity model and the number of the gravity terms. Figure 3(a) shows the variation in periapsis of a 100 km circular polar lunar orbit using 10 × 0, 10 × 10, 15 × 15, 20 × 20, 50 × 0 and 50 × 50 terms of the lunar gravity model LP150Q. Vast difference is seen in lifetime estimated by (10 × 10) and (50 × 50) terms. Assessment from Figure 3(b) shows that the lifetime estimated using 50 × 50 and 60 × 60 terms are comparable.

A comparison of lifetime estimated with 5 lunar gravity models (LP75G, LP100K, LP100J, LP165P, and LP150Q) using 50 × 50 gravity terms for a circular polar lunar orbit of 125 km altitude is shown in Figure 4. It can be observed that the lifetime calculated with the models LP75G and LP100 K is around 384 days while that computed with the models LP100J, LP165P and LP150Q is 411 days.

(a)(b)

Figure 3. (a). Variation of periapsis of a 100 km circular polar lunar orbit using different gravity terms of lp150q; (b). Variation of periapsis of a 100 km circular polar lunar orbit using 50 × 50 and 60 × 60 terms.

Table 4 shows the comparison of lifetime of lunar orbits estimated with 50 × 50 gravity terms and with 50 zonal harmonics terms of LP150Q model without including Sun and Earth point mass gravity effect. It can be observed that the orbital lifetime predicted using only zonal harmonics gives higher values as compared to the orbital lifetime computed with 50 zonal and 50 tesseral terms in most of the cases. This indicates the significance of tesseral harmonics terms in lunar gravity model for assessing the orbital lifetimes of lunar satellites.

It is noticed that for a fixed value of Ω, the lifetime increases significantly after a particular altitude. This is essentially due to the gradual reduction of the effect of the long periodic terms of Moon’s gravity on eccentricity of the orbit. It is depicted from Figures 5(a) and 5(b)

Figure 4. Variation of periapsis for last 31 days of a 125 km circular orbit using different lunar gravity model.

(a)(b)

Figure 5. (a) Evolution of eccentricity of a 138 km circular polar orbit (Ω = 25˚); (b) Evolution of eccentricity of a 99 km circular polar orbit (Ω = 125˚).

where the period of the long periodic term on eccentricity for two orbits with initial altitude 138 km (Ω = 25˚) and 99 km (Ω = 125˚) are 920 days and 870 days, respectively. It may be noted that the long periodic terms due to 50 × 50 Moon’s gravity have different period of long periodic terms on eccentricity than the 50 × 0 Moon’s gravity which may be seen in Figures 5(a) and 5(b). Initially the periapsis starts decreasing and if it does not reaches to zero altitude, it again starts increasing. Then it has to take at least one cycle to reach to zero altitude resulting in higher orbital lifetime. As observed in Figure 6, for the case where Ω = 60˚ and the initial altitude of 130.25 km, the periapsis becomes zero after 414 days. In Figure 7, for initial altitude of 130.50 km, it is observed that the periapsis first starts decreasing and has periapsis altitude slightly greater than zero and then again starts increasing. Thus, it leads to a higher orbital lifetime.

Figure 6. Periapsis variation of a circular polar orbit with initial altitude 130.25 km and Ω = 60˚.

Figure 7. Periapsis variation of a circular polar orbit with initial altitude 130.50 km and Ω = 60˚.

Depending on the initial altitude, the altitude of the orbit can be zero in the first cycle itself, if not, the periapsis increases due to long periodic terms and comes down in next cycle and can become zero. Hence for a particular initial altitude, periapsis can vanish in its first cycle while for a little higher altitude it lifts and lifetime can be much higher. We define the transition altitude as the altitude for which the orbital lifetime increases substantially (at least one cycle more) with a very small increase (of the order of 0.1 to 0.2 m) in the altitude. Figure 8 shows the variation of transition altitude with Ω for circular polar orbit. The transition altitude is different for different values of Ω and follows a definite trend. The plot also shows the difference in transition altitude when Sun and Earth gravity are ignored. It is observed from Figure 8 that for circular polar orbits the transition altitude computed without considering Sun and Earth gravity is more than that computed using Sun and Earth gravity effect. It is essentially due to the effect of the long periodic terms of Earth’s gravity on the eccentricity of the lunar satellite orbit.

On considering Sun and Earth gravity effects for circular polar orbits, the transition altitude decreases from 138 km (Ω = 30˚) to 99 km (Ω = 120˚) and again in-

Figure 8. Variation of transition altitude with Ω for circular polar orbit.

creases to 136 km (Ω = 210˚) and then decreases to 114 km (Ω = 290˚). It can be seen that for circular polar lunar orbit, for a given Ω, the lifetime will be of the order of 300 days to 400 days for the altitudes below this curve, while for the altitudes above this curve lifetime will be of the order of 3 years or more. As a particular case, Figure 9 shows the variation of orbital lifetime with altitude for a circular polar orbit with Ω = 0˚. Table 3 gives the lifetime of circular polar orbit at transition altitudes.

Figure 10 shows the periapsis variation for an orbit with an initial altitude of 140 km for 10,000 days. From this figure, it can be inferred that if the long periodic terms due to Moon’s gravity on eccentricity are not able to bring down the periapsis to zero then the lifetime will be very long.

Figures 11, 12 and 13 show the variation in transition altitude for circular orbits with initial orbital inclination 10˚, 50˚ and 70˚^{ }and the transition altitude varies from 138 km to 182 km, 40 km to 156 km, and 33 km to 270 km, respectively. It may be noted that different trends in transition altitudes are seen for different inclinations with variation of Ω from 0 to 360^{ }degrees. The effect of altitude on lifetime of circular lunar orbit of different incli

Table 3. Lifetime of circular polar orbits at transition altitudes.

Table 4. Comparison of lifetime of lunar polar orbit estimated using 50 × 50 and 50 × 0 lunar force model.

nation is also analyzed for Ω = 0˚. Figure 14 shows the variation of transition altitude with orbital inclination. For all the results presented here for circular orbits, the argument of periapsis and the true anomaly are taken as zero for the initial conditions. Ω is varied from 0˚ to 360˚ in the intervals of 1˚ to 10˚ depending upon the trend in variation. For each value of Ω, the lifetime is calculated for different altitudes ranging from 30 to 500 km.

Figures 15(a) to 15(f) show the variation in orbital parameters of a 130 km altitude circular polar orbit. As seen from Figure 15(d) Variation of nearly 2 degrees in inclination is noticed.

3.3. Lifetime of High Altitude Lunar Orbits

Apart from the gravity field of Moon, lifetime of high altitude lunar orbit is mostly influenced by gravitational field of Earth. At higher altitudes, the perturbations due to Earth’s gravity have significant effect on the lifetime of lunar orbits. The variation of periapsis of 1200 km circular orbit (inclination = 55˚) without considering Earth and Sun gravity, considering the Earth’s gravity alone and considering both Sun and Earth gravity is given in Figure 16(a). Figure 16(b) gives the variation in eccentricity of the orbit. It can be seen that gravity effect of Sun causes very less variation on the periapsis variation and hence on the orbital lifetime. The perigee variation shows a very high lifetime if the gravity effect of Sun and Earth are not considered. Hence for the lifetime estimation of lunar orbits, the gravity effect of Earth is essential.

Figures 17(a), 17(b), 17(c) and 17(d) give the variation in lifetime with altitude for circular orbits with inclinations of 60˚, 65˚, 63.4˚ and 90˚. For an orbit with

Figure 9. Variation of orbital lifetime with altitude for Ω = 0˚.

Figure 10. Periapsis variation of a 140 km altitude circular orbit during 10,000 days.

Figure 11. Variation of transition altitude with Ω (inclination = 10˚).

Figure 12. Variation of transition altitude with Ω (inclination = 50˚).

Figure 13. Variation of transition altitude with Ω (inclination = 70˚).

Figure 14. Variation of transition altitude with orbital inclination.

(a)(b)(c)(d)(e)(f)

Figure 15. Variation in Orbital Parameter of a 130 km initial altitude polar orbit (Initial Ω = 0˚). (a) Periapsis variation; (b). Apoapsis variation; (c). Variation in eccentricity; (d). Variation in inclination; (e). Variation in argument of periapsis; (f). Variation in Ω.

(a)(b)

Figure 16. (a). Variation of periapsis of 1200 km circular orbit (i = 55˚); (b). Variation of eccentricity of 1200 km circular orbit (i = 55˚).

(a)(b)(c)(d)

Figure 17. (a). Variation of lifetime with altitude for circular orbit (i = 60˚); (b). Variation of lifetime with altitude for circular orbit (i = 65˚); (c). Variation of lifetime with altitude for circular orbit (i = 63.4˚); (d). Variation of lifetime with altitude for circular orbit (i = 90˚).

inclination 60˚, the altitude is varied from 100 km to 6500 km (cf. Figure 17(a)). The lifetime first increases from 83 days (for 100 km orbit) and becomes maximum (5197 days) at an altitude of 1800 km and then it starts decreasing. At an altitude of 6500 km, the lifetime becomes 845 days. In Figure 17(b), altitude is varied from 1000 km to 9000 km. The lifetime first increases and it is maximum at 1500 km altitude and then starts decreasing. Same trend in lifetime variation can be observed in Figure 17(c). For circular polar orbit the altitude is varied from 1200 km to 9000 km. The variation of lifetime with altitude for circular polar orbit is shown in Figure 17(d).

Lifetime of high altitude circular lunar orbits decreases with altitude. This is due to the increase of Earth’s gravitational effect on long periodic terms in eccentricity i.e. the eccentricity starts increasing resulting in the decrease of perigee altitude to zero.

4. Conclusions

The effect of right ascension of ascending node (Ω) on the variation of lifetime with altitude is assessed. It is found that for a given Ω, the lifetime increases with altitude and at a particular altitude defined as transition altitude lifetime increases substantially due to the effect of long periodic terms of Earth’s gravity (zonal and tesseral harmonics) on eccentricity of the orbit. Earth’s gravity dominates over the Moon’s gravity with respect to the long periodic effect on eccentricity of the orbit. The transition altitude varies from 138 km to 182 km, 40 km to 156 km, 33 km to 270 km and 99 km to 138 km for orbits with inclination of 10˚, 50˚, 70˚ and 90˚, respectively. There is significant effect of Earth gravity on transition altitudes while the effect of Sun’s gravity is very marginal. Variation of transition altitude with orbital inclination is also analyzed. Lifetime of high altitude circular lunar orbits is analyzed and it is observed that at high altitudes lifetime decreases with altitude.

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