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How to get to the Moon: 2023 edition

Updated: Dec 29, 2023




It was definitely a busy year when it comes to lunar missions... With three launches in a period of approximately two months, the ISRO, JAXA and Roscosmos sent three probes towards the Moon: Chandrayaan-3, LUNA-25 and SLIM.


Although all three missions had a common destination, their itineraries were vastly different from each other. Since the SLIM probe just executed its Lunar Orbital Insertion (LOI) maneuver (25th of December 2023), I figured it would be a good opportunity to take a look at the different trajectories chosen by each of the mission design teams, outline some of the factors that played a role in these decisions and to try and loosely reproduce the chosen trajectories given the available information on the web.


Mission description


LUNA-25

The Luna-25 lander was the first Soviet or Russian Moon probe since Luna-24 in 1976. With that, Luna-25 signified Russia’s resurgence in lunar exploration, marking its return after a 47-year hiatus and aiming to reestablish its legacy in the realm of space exploration. The mission plan for the probe was to launch atop a Soyuz-2.1b rocket, and after a 5-day trip to the moon to circle the natural satellite for another five to seven days. The plan for the spacecraft was to then set down in the moon's south polar region, near Boguslawsky Crater. Unfortunately, the probe crashed into the moon during a maneuver designed to set up a touchdown attempt near the lunar south pole. It turns out that an onboard control unit failed to turn the engines off because it wasn't receiving the necessary data from one of Luna-25's accelerometers according to this source.


Chandrayaan-3

Chandrayaan-3 is a mission developed by the Indian Space Research Organisation that was launched on the 14th of July 2023, consisting of an orbiter, a lunar lander and a lunar rover. It represented India’s third lunar mission and its second endeavor to achieve a gentle lunar surface landing. The objectives of the mission were to safely and softly place a lander on the surface of the Moon, to guide the rover and to carry out the characterization of the mineralogical composition of the lunar surface and the elemental composition of its rocks. The lander successfully touched down near the lunar south pole of the Moon on 23 August 2023 (source).


SLIM

The Smart Lander for Investigating Moon (SLIM) is Japan's first lunar surface mission, and aims to demonstrate precise, pinpoint lunar landing. During its descent to the Moon, the lander will recognize lunar craters by applying technology from facial recognition systems and determine its current location from utilizing observation data collected by previous lunar orbiter missions like SELENE. The main objective for SLIM is to soft land with an accuracy range of 100 m, which is orders of magnitude smaller than the 5km x 20km landing ellipse of the Apollo 11 Eagle lunar module. On the 25th of December 2023, SLIM successfully entered a lunar orbit (source).


Due to their launch dates being so close to each other and the fact that they both attempted a landing on the Moon's south pole, several articles have emerged trying to compare LUNA-25 and Chandrayaan-3 (e.g. this one). Instead of going into the missions objectives in detail, I would like to share some insights on their respective trajectories and how they can be modelled.


Trajectories comparison


I tried summarizing the initial (Earth-bound) and target (Moon-bound) orbits for each one of the probes along with the maneuvers that they carried out along the way.

Due to the lack of exact published information for the inclination of the Earth-bound orbits for Chandrayaan-3 and SLIM, I used the two-line elements (TLE) reported by NORAD artificial satellite tracking in https://www.space-track.org/.

Mission

Starting orbit

(Earth-bound)

Target Orbit

(Moon-bound)

Maneuvers

​LUNA-25

​​267km x 281km

52deg incl. [S]

​91km x 112km

82deg incl. [S]

​​1) Trans-lunar injection TLI (using Fregat upper stage)

2) Lunar-orbit injection LOI (using on-board engines)

​Chandrayaan-3

​170km x 36500km

21deg incl. [S1, S2]

​153km x 163km

85deg incl. [S]

​​1) 5x orbit-raising maneuvers

2) TLI

3) LOI

4) 4x moon-bound maneuvers

SLIM

​​581km x 112,237km

31deg incl. [S]

​​15km x 600km

85deg incl. [S]

​​1) TLI

2) Un-powered moon flyby

3) Deep-space maneuver DSM

4) LOI


LUNA-25

Launching with a Soyuz-2.1b with a C3 energy of ~7 km**2/s**2 (Source from 2018), LUNA-25 mission designers chose a direct path to the moon. After an initial phase of staying on a nearly circular parking orbit with ~280 km altitude, the Fregat upper stage ignited once more to place the spacecraft onto a lunar transfer orbit. Given the fact that the launch was done using such a powerful launch vehicle, the requirements for the on-board propellant mass of the LUNA-25 probe were significantly reduced: after the trans-lunar injection (TLI), only two correction maneuvers had to be carried out by the on-board propulsion system, before the final lunar transfer orbit insertion and the landing maneuver.


Luna-25 mission profile. Credit: Roscosmos / Lavoichkin

Modeling

The following assumptions and approach was taken to simulate the LUNA-25 mission:

  • Starting Earth-bound orbit of 52deg incl. @ 267km x 281km

  • Target Moon-bound orbit of 82deg incl. @ 91km x 112 km

  • Objective: minimization of the total DeltaV (sum of the TLI carried out by the Fregat upper stage and the LOI carried out by the LUNA-25)

  • Free optimization parameters:

  1. Right ascension of the ascending node (RAAN) & argument of perigee of the Earth-bound orbit

  2. x/y/z components of the TLI and LOI DeltaVs

  3. Journey duration between the TLI and the LOI

  • The descent maneuver from the low lunar orbit (LLO) to the surface of the Moon was not included in this simulation.

  • Matlab's fmincon function with the Interior Point Algorithm was used for the optimization

  • The circular restricted three-body problem (3D) was used for the dynamics

The results of the optimization are seen in the following animation. With the current constraints, a travel duration of approximately 5 days was achieved, leading to a total DeltaV consumption of slightly under 4 km/s. At the end of the simulation, the orbiter keeps traveling along its LLO for a couple more days, which deviates from the actual mission profile, but was chosen for better visualization of the trajectory's geometry.




LUNA-25 trajectory as seen in the inertial frame.
LUNA-25 trajectory as seen in the rotational frame of the Earth-Moon system.

Chandrayaan-3

Looking into the mission profile of the ISRO's moon mission, one immediately identifies that the chosen trajectory is quite different from Roscosmo's design. Chandrayaan-3 launched with a Launch Vehicle Mark (LVM3) and was directly placed into an elliptical orbit around the Earth (altitude of 170km x 36,500km). After being placed into this highly elliptical orbit, it kept revolving for a day until a series of successive orbit raising maneuvers were executed by the propulsion module within a span of 10 days. The final elliptical orbit before the TLI was a 236 km x 127,600 km orbit and after the TLI maneuver, the probe traveled for approximately 6 days before reaching the Moon. Once again with the help of the propulsion module, the LOI was executed, placing the probe into a 164 km x 18,000 km elliptical lunar orbit and was followed by 4 further lunar bound maneuvers before reaching a nearly circular low lunar orbit (LLO) with altitude 153 km x 163 km 12 days after the LOI. After that point, the landing module (Vikram) was detached from the propulsion module before starting the de-orbiting and landing maneuvers.


The chosen trajectory reflects the principle of frugal innovations that India's space program is based on, according to which the missions are made as cost-effective as possible without compromising the scientific objectives. For that reason, and considering the lower lifting capacity of the LVM3, a longer and very fuel-efficient journey to the moon was chosen.

A similar mission design methodology was followed in two of the preceding moon missions, Chandrayaan-1 in 2008 and Chandrayaan-2 in 2019.


Schematic design of the Chandrayaan-3 mission. Source: ISRO

Modeling

The following assumptions and approach was taken to simulate the Chandrayaan-3 mission:

  • Starting Earth-bound orbit of 21.33 deg incl. @ 170 km x 36,500 km

  • Target Moon-bound orbit of 85 deg incl. @ 153 km x 163 km

  • Objective: minimization of the total DeltaV (sum of the 5x orbit-raising maneuvers, the TLI, the LOI and the 4x lunar-bound maneuvers)

  • Free optimization parameters:

  1. Right ascension of the ascending node (RAAN) & argument of perigee of the initial Earth-bound orbit

  2. x/y/z components of all DeltaVs

  3. Journey duration between the TLI and the LOI and all time durations between the successive Earth-bound orbital raises and Moon-bound de-orbiting maneuvers.

  • The descent maneuver from the low lunar orbit (LLO) to the surface of the Moon was not included in this simulation.

  • Matlab's fmincon function with the Interior Point Algorithm was used for the optimization

  • The circular restricted three-body problem (3D) was used for the dynamics

The results of the optimization are seen in the following animation. With the current constraints, a travel duration of approximately 29 days was achieved, leading to a total DeltaV consumption of slightly under 1.7 km/s.





Chandrayaan-3 trajectory as seen in the inertial frame.
Chandrayaan-3 trajectory as seen in the Earth-Moon rotational frame.


SLIM

The SLIM spacecraft launched on September 6, 2023 on board an H-IIA launcher and was initially placed onto a highly elliptical orbit around the Earth, with an apogee altitude of approximately 112,000km. All subsequent orbital maneuvers were carried out using the pair of 500 N on-board ceramic engines. Initially, JAXA announced that the total duration of the journey to the Moon would take between 4 and 6 months and indeed SLIM entered its starting lunar orbit on the 25th of December, i.e. 110 days after launch. The reason for this prolonged journey duration compared to Chandrayaan-3 and LUNA-25 is fuel-efficiency. The SLIM missions is (as its name suggests) a stripped down mission with only a couple of miniaturized scientific payloads on board, as the primary objective is to execute a precision landing. After its TLI on the 30th of September 2023, the probe utilized a lunar fly-by on the 4th of October, in order to bring its velocity vector in a favorable orientation for its second encounter with the Moon. The chosen mission profile brought the probe to the region outside of the Moon's orbit, before it performed an impulsive maneuver and phased itself towards its target destination.



Modeling

The following assumptions and approach was taken to simulate the SLIM mission:

  • Starting Earth-bound orbit of 31 deg incl. @ 581 km x 112,237 km

  • Target Moon-bound orbit of 85 deg incl. @ 15 km x 600 km

  • Objective: minimization of the total DeltaV (sum of the TLI, the LOI and the mid-course maneuver)

  • Free optimization parameters:

  1. Right ascension of the ascending node (RAAN) & argument of perigee of the initial Earth-bound orbit

  2. x/y/z components of all 3 DeltaVs (TLI, mid-course, LOI)

  3. Journey duration between the TLI and the first lunar flyby, the duration between the flyby and the mid-course maneuver as well as the duration between the mid-course maneuver and the LOI.

  • The altitude of the first lunar flyby was constrained to be between 100km and 1000km.

  • The descent maneuver from the low lunar orbit (LLO) to the surface of the Moon was not included in this simulation.

  • Matlab's fmincon function with the Interior Point Algorithm was used for the optimization

  • The circular restricted three-body problem (3D) was used for the dynamics

  • Note that in order to properly account for the dynamics of the SLIM probe, the force from the Sun (as a fourth-body perturbation) should be accounted. This was not done in this simulation as it does not affect the general mission profile.

The results of the optimization are seen in the following animation. With the current constraints, a travel duration of approximately 65 days was achieved, leading to a total DeltaV consumption of slightly under 1.2 km/s. The optimal solution predicted by this model delivers a maximal distance of approximately 1.4 million km from the moon and a maximal distance of ~650,000 km form the Earth, i.e. roughly double the distance of the moon's orbital semi-major axis.



SLIM trajectory as seen in the inertial frame.

SLIM trajectory as seen in the Earth-Moon rotational frame.

In order to make the comparison more direct, the following animation shows all three missions synchronized with their arrival time into their target lunar orbit. For the Chandrayaan-3 mission, the number of loops around the Earth before the TLI was reduced in the to make the animation less busy (and since it does not change the total DeltaV for the mission significantly).




Obviously, for each of the three mission profiles, the simulations are a loose interpretation of the actual executed missions, due to the lack of precise information on some of the constraints, boundary and initial conditions. Of course the simplified objective of the optimization (total DeltaV) is also not 100% accurate, as for real missions several other constraints have to be traded off, which are not reflected in this simple problem formulation. Nevertheless, I still hope that they give a representative overview of the differences between the approaches as well as a realistic estimate for the expected fuel costs and associated travel durations.




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