## Starship Trajectory for a 2024 Mars Mission

Mars could be three years away if we choose it to be. We will explore two possible trajectories for a SpaceX Starship 2024 Mars mission. I choose SpaceX’s Starship to design, calculate and plot the trip trajectories, from launch to Landing, as its specs are currently the most promising and viable option available to date.

- In
**Part 1**, we will explore a detailed Earth to Mars path based on a heliocentric Hohmann Transfer Orbit path. - In
**Part 2**, based on my calculations, I was able to reduce the total mission time (Launch to Landing) from 11 months to 6 months and be a viable mission. So, I will present the updated values and explore what additional maneuvers will be needed to accomplish the mission’s success.

## Part 1

Proposal for a 2024 Mars Mission following a Hohmann Orbital Transfer trajectory; ( a 9 months Heliocentric transfer & 11 months total trip time**)**

### Starship’s Launch & Powered Flight to Orbit

In early October 2024, on Ramp 39A of the Kennedy Space Center, Florida, Spacex‘s Starship (SS) securely mounts the Super Heavy Rocket(SHR), fully fueled and counting down for launch. It carries around 1,200 tons of fuel and more than 100 tons of payload. The Super Heavy rocket has a fuel capacity of 3,400 tons. Together, they have a combined lift-off mass of 5,300 Tons.

The window for launch is narrow and closing. On this launch day, Earth is 109 days from Opposition with Mars (January 16th, 2025) and 33 days from reaching the 40.242 degrees phased angle with Mars.

Seconds before the launch, the Super Heavy Rocket engines ignite. At T-0, The Rocket computer signals the mount release, and the SHR carrying the Starship lifts off. At a height between 11 – 13 km, the rocket reaches the maximum aerodynamic pressure max Q. The SHR continues on a powered flight until it reaches an altitude between 80 – 100 km. It then initiates the first stage main-engines cutoff (MECO). A few seconds after, the HSR separates from SS and starts reentry, and then lands vertically on an offshore landing pad.

### Parking the Starship in Earth Orbit

Few seconds after the Starship separates from the SHR; it initiates engine ignition. It then goes through a powered flight towards the planned parking orbit (PO) of 200km (108 n. miles). SS applies the Parking orbit Maneuver Injection (POMI) to reach the designated altitude. POMI allows the Starship to course correct and align with the target Circular Orbit(CO). It then continues its powered flight until it reaches the required Circular Speed(CS) v_{cs} = 7.784 km/sec required to maintain a 200 km CO. As the ship approaches CS, its engines will engage in engine cutoff. The Starship will then coast to orbit and start orbit verification.

### Refueling Starship

Once parked in orbit, the Starship starts a system-wide status check (SSC). Since it has exhausted its fuel during the second stage burn, SpaceX has come up with a plan to refuel the Starship in orbit. So, it undergoes a Guided System Reference Alignment (GSRA) in preparation to rendezvous with the refueling Tanker-ship.

The Tanker-ship (TS) will have either been in the parking orbit waiting or just followed the Starship and achieved PO shortly after. When both ships are in alignment, the Starship will rendezvous and connect with TS and start the refueling process. Once refueling is complete, the Starship disengages, and the Tanker returns to Earth and lands. As the Starship approaches the Trans-Mars Injection (TMI) point, it starts the last system update and System Status Check. The positive system checks and proper alignment warrants a “Go” Decision.

### Starship in Earth Geocentric Trajectory

The Starship approaches the TMI coordinate on its parking orbit. At the perigee of the hyperbola, where the angle between Earth’s orbital velocity vector and the injection radius(r_{0}=6578km) vector is η_{1}=149.18 degrees, the Starship ignites its engines and initiates TMI-0. SS then goes through a powered flight until it achieves a velocity v_{0}= 11.45235 km/sec.

Velocity v_{0} is required to escape the Earth sphere of influence (ESoI) and still have the right amount of velocity to coast the Heliocentric transfer orbit towards Mars. Once v_{0} is achieved, SS’s engines cut off, and it slingshots coasting along a hyperbolic trajectory until it exits the ESoI. The hyperbolic orbital trajectory has a semi-major axis a_{0}=40003.16852 km, an eccentricity e_{0}=1.1644369, with Earth’s center as the hyperbola’s focus, a specific angular momentum ε_{t0} = 4.98212 km^{2}/sec^{2}, an eccentric anomaly F_{0}=212.4719917 Degrees and an angle between the asymptotes of the hyperbola Delta δ_{1} = 118.360677 Degrees.

### Starship Approaches ESoI

As the Starship coasts the hyperbolic trajectory, it does periodic system checks and navigation sightings. It approaches the ESoI, 29.369 days later. SS now has a hyperbolic excess speed ∆v = 3.1566185 km/sec relative to Earth. Since Earth’s orbital speed is 29.78 km/sec, the Starship’s velocity, relative to the Sun, is V_{t1} = 32.9366185 km/sec. (t1) is the time the Starship starts its heliocentric transfer orbit to Mars. At the edge of the ESoI, SS undergoes a TMI-1 to course-correct as it commences its heliocentric orbit. It might also need several consecutive corrective TMIs if needed.

### Starship’s Window of Launch

The Starship has a limit window to achieve a fuel-efficient Hohmann Transfer Orbit. When it starts the heliocentric elliptic transfer orbit, SS will be at a phased angle γ_{1} = 40.24223 Degrees from Mars. At this phased angle, it would have been 75.855 Degrees from the 2025 Opposition with Mars ( the January 16th); and an eccentric anomaly E=180 Degrees.

### Starship in Heliocentric Hohmann Transfer Orbit

The heliocentric Hohmann transfer orbital trajectory to Mars has a semi-major axis a_{ho}=1.92493567 x 10^{8} km, an eccentricity e_{ho}=0.22283299 with Sun’s center as the hyperbola’s focus r_{t1} = 1 AU, a specific angular momentum |ε_{t1}| = 344.7269487 km^{2}/sec^{2} and a semi-minor axis of b_{ho} = 1.87653631 x 10^{8} km.

The Starship cruises along the precalculated heliocentric orbital trajectory and undergo periodic navigation sighting, system-wide checks & computer updates, and guidance system reference alignments.

The Earth’s orbital plane around the Sun is at an angle i=1.85 degrees from that of Mars. As the Starship approaches a designated point on the trajectory (r_{i}= 1.92493567 x 10^{8} km from the Sun), it undergoes attitude orientation and a midcourse planer correction. At the point r_{i} on the trajectory, SS ignites its engines for a short burn. It then applies a TMI-3 inducing a velocity ∆v_{ti}= 0.8477779373 km/sec. The velocity is enough to maintain the speed from before the inject and carry it to after the injection to support the specific angular momentum along the Heliocentric coasting.

### The Starship Approaches Mars Sphere of Influence (MSoI)

SS approaches the Mars Sphere of Influence (MSoI = 5.78 x 10^{5} km) after a heliocentric travel time of t_{2} = 266.5629 days (8.67 months). Upon arriving at the MSoI with an Offset Y_{2}=5.8123 x 10^{3} km from Mars’s orbital velocity vector and a Velocity v_{t2} = 20.9327468 km/sec, the SS induces TMI-4 to course-correct and change from a Heliocentric Model to a Mars Geocentric Model. Additional TMIs will be initiated as needed to achieve the correct Mars geocentric entry. If Mars’s orbit around the Sun were circular, the Starship would approach Mars orbit tangentially. However, since Mars’s orbit is elliptic, SS will approach the MSoI with an angle Ф_{2}=5.05 Degrees. It will start its Mars geocentric hyperbolic approach at a Mars relative velocity of v_{m1}= 3.708637927 km/sec.

### Starship Coasts a Hyperbolic Mars Intercept Orbit

SS starts cruising on a hyperbolic trajectory to undergo an aerobraking maneuver at 100 km above Mars’s surface. The Mars reentry hyperbolic orbital trajectory has a semi-major axis a_{m1} = 3113.858856 km, an eccentricity e_{m1}=2.117584374, with Mar’s center as the hyperbola’s focus r_{o2} = 3.380 x 10^{3} km, a specific angular momentum |ε_{t2}| = 6.8769976 km^{2}/sec^{2}, an eccentric anomaly F_{m1}=295.8416269 Degrees and an angle between the asymptotes of the hyperbola Delt δ_{2} = 56.3591 Degrees.

The Starship approaches the hyperbolic perigee r_{o2} = 3.380 x 10^{3} km. At the perigee, it has an angle between Mars’s orbital velocity vector and the perigee radius vector of η_{2}=113.129554 degrees and velocity v_{o2}=6.1941735 km/sec.

### Starship Approaches Mars

SpaceX claims that its Starship, as it undergoes Mars altitude penetration, will withstand structurally and the heat from the atmosphere drag while keeping its payload and crew safe as it enters the atmosphere at a 7.5 km/sec velocity.

Since v_{o2} is less than the acceptable safe entry velocity proposed by SpaceX, Starship has two options: Land or Park in Orbit. In this case, due to a sand storm over the landing site, the decision is to park in orbit till the storm clears. The velocity of the Starship, to achieve a stable circular orbit at 100km from Mars’s surface, should be v_{cs2}=3.5081 km/sec. At velocity v_{o2}=6.1941735 km/sec SS comes to the end of its reentry hyperbolic trajectory. Now it heads towards Mars for an atmosphere reentry.

### Starship First Walk-in for Atmospheric Aerobraking

At the hyperbolic perigee, 100 km from Mars surface SS experience high drag caused by Mars atmosphere. This aerobraking caused by atmospheric drag shaves a considerable amount of its velocity. The ship exists the Martian atmosphere, with a velocity of v_{m2}=4.794172 km/sec. It puts it on an elliptic orbit with Mars as its focus where the semi-major axis a_{m2}=2.6279825 x 10^{4} km, an eccentricity e_{m2}= 0.867579, a specific angular momentum |ε_{m2}| = 0.8148456 km^{2}/sec^{2}.

The Starship needs 14.97 days to complete the orbit and be back where it started at the ellipse’s perigee. During this orbit, SS will perform regular system checks and reference alignments and monitor Mars’s condition for atmospheric reentry.

### Starship Parks in Orbit Around Mars

The conditions are not right so the Starship does another aerobraking maneuver. This time exiting the atmosphere at v_{m3}= 3.794173 km/sec. This maneuver will put it on a one-day elliptic orbit. This orbit has a semi-major axis a_{m3}=4.191411 x 10^{4} km, an eccentricity e_{m3}=0.16973, and a specific angular momentum |ε_{m3}| = 5.10910935 km^{2}/sec^{2}. After a second orbit, SS does another orbital parking maneuver using the velocity decay and reentry drag. After this maneuver, it will have a velocity v_{m4} = 3.5081186 km/sec. That velocity allows a circular stable parking orbit at 100 km altitude from Mars’s surface. SS can not just jump directly to a stable orbit. It will need to make several orbits applying minor maneuvers to fall into the stable circular parking orbit.

### Starship Lands on Mars

When the Starship’s and Mars’s conditions are optimal, SS will position itself and start the atmospheric reentry. It starts the Mars Entry Maneuver Injection (MEMI), then does a free fall atmospheric entry. At around 15km altitude the Starship positions itself into the Horizontal Mode. Its raptor engines then ignite and initiate the entry burn, then proceeds with the aerodynamic guidance, and finally, land vertically on the surface of Mars.

## Part 2

The Hohmann-based trajectory illustrated in Part1 has a very long total flight time. This flight time is long partially because a 2024 mission, the Starship, will coast in the Mars aphelion zone, where Mars is the farthest from the sun. The flight time for the same trajectory design approach will be significantly less for a 2028 Mars mission; where the transfer orbit is in the Mars parhelion zone.

### Revised flight timeline for the Starship mission

There are many significant reasons for a shorter Earth to Mars trip that are critical to the success of the mission. Among the reasons are the effects of prolonged exposure to radiation and microgravity on the health of the astronauts. Also, there is mission operation and cost challenge considerations.

In this scenario, the total flight time from launch to landing is reduced to six (6) months. This new parameter has a cascade effect on the type of trajectories and values for the entire mission.

One of the main consequential changes is that the Hohmann transfer orbit does no longer applies. Instead, a new elliptical transfer orbit is required to achieve the heliocentric orbit.

### Starship’s Window of Launch

When the Starship starts the new Heliocentric elliptic transfer orbit, SS will be at a new phased angle γ_{1} = 36.605634 Degrees from Mars. At this new Phased angle, it would have been 69 Degrees from the 2025 Opposition with Mars.

### Refueling and Boosting the Starship

Just like in Part 1, the Starship parks in orbit and refuels from the Tanker Ship. Then once refueling is complete, the Starship disengages. However, the Ship needs a velocity v_{0}= 12.19234301 km/sec. A 2 km/sec increase from before. This increase will also result in a much higher mars approach speed; it will require a burn maneuver to reduce the approach speed.

So, in reality, it means the Starship will need much more fuel. A fuel volume that it will not have the capacity to carry unless SpaceX builds a bigger model. The solution is for the starship to mount a booster rocket in orbit. It will use the same procedure of connection as that with the Fueling Tanker. Then, as it approaches the Trans-Mars Injection (TMI) point the booster rocket initiates an injection burn. This Burn propels the Starship on a hyperbolic trajectory.

### Starship in Earth Geocentric Trajectory

The Booster injection burn, TMI-0, is initiated at the perigee of the hyperbola, where the angle between Earth’s orbital velocity vector and the injection radius vector is η_{1}=133.4834856 degrees. The Booster carries the Starship through a powered flight until it achieves a velocity v_{0}= 12.19234301 km/sec.

Velocity v_{0} is required to escape the Earth’s Sphere of Influence (ESoI) and still have the right amount of velocity to coast the heliocentric transfer orbit towards Mars. Once v_{0} is achieved, the booster ships’ engines cut off, disengages from the starship and returns to earth. The Starship slingshoots coasting along a hyperbolic trajectory until it exits the ESoI. The hyperbolic orbital trajectory has a semi-major axis a_{0}=14515.16817 km, an eccentricity e_{0}=1.453181108, with Earth’s center as the hyperbola’s focus, a specific angular momentum ε_{t0} = 13.73050575 km^{2}/sec^{2}, an eccentric anomaly F_{0}=261.7033509 Degrees and an angle between the asymptotes of the hyperbola Delta δ_{1} =86.96697123 Degrees.

### Starship Approaches ESoI

As the Starship coasts the hyperbolic trajectory, it does periodic system checks and navigation sightings. The Starship approaches the ESoI, 20.967 days later. The ship now has a hyperbolic excess speed ∆v = 5.240325514 km/sec relative to Earth. Since Earth’s orbital speed is 29.78 km/sec, the ship’s velocity, relative to the Sun, is V_{t1} = 35.02032551 km/sec. At the edge of the ESoI SS undergoes a TMI-1 to course-correct as it commences its Heliocentric Orbit. It might also need several consecutive corrective TMIs if needed.

### Starship in Heliocentric Elleptic Transfer Orbit

The new Heliocentric elliptic orbital trajectory to Mars has a semi-major axis a_{ho}=2.422470875 x 10^{8} km, an eccentricity e_{ho}=0.382450161 with Sun’s center as the hyperbola’s focus r_{t1} = 1 AU. The specific angular momentum is |ε_{t1}| = 273.9257701 km^{2}/sec^{2} and the semi-minor axis is b_{ho} = 2.238305246 x 10^{8} km.

Just like in the Hohmann option, the Starship cruises along the pre-calculated heliocentric orbital trajectory. It undergoes periodic navigation sighting, system-wide checks & computer updates, and guidance system reference alignments.

As the Starship approaches the designated point on the trajectory (r_{i}= 1.92493567 x 10^{8} km from the Sun), it undergoes attitude orientation and a midcourse planer correction. At the point r_{i} on the trajectory, the Starship ignites its engines for a short burn. It then applies a TMI-3 inducing a velocity ∆v_{ti}= 0.930776568 km/sec. The velocity is enough to maintain the speed from before the inject and carry it to after the injection to support the specific angular momentum along the Heliocentric coasting.

### The Starship Approaches Mars Sphere of Influence (MSoI)

SS approaches the Mars Sphere of Influence after a heliocentric travel time of t_{2} = 151.963 days (5 months). It’s new Velocity v_{t2} = 22.76271117 km/sec and has an angle Ф_{2}=22.91 Degrees. This is significant because the Mars relative velocity will be v_{m1}= 9.388728876 km/sec. That results in a 5.68 km/sec difference from that of the Hohman transfer. This additional speed will result in a Mars catching speed that causes the ship not to catch orbit. The aerobraking maneuver used in the first Hohmann option will not be enough to reduce the speed. A speed needed to achieve a walk-in into an elliptic transfer orbit.

## Starship Applies Early Reverse Burn Maneuver.

The only course of action is to induce TMI-4 first to reduce the velocity to a manageable speed. Also, to course-correct and change from a Heliocentric Model to a Mars Geocentric Model. The Starship does a 180 turn Maneuver, followed by a reverse Burn injection to slow it down. This burn will shave 5.5 Km/sec of its original approach velocity. After the burn, the engines cut off and SS does another 180 turn maneuver orienting to Mars. The new Mars relative speed is now v_{m1}= 3.888728876 km/sec. Additional TMIs will be initiated as needed to achieve the correct Mars geocentric entry.

### Starship Coasts a Hyperbolic Mars Intercept Orbit

The Starship starts cruising on a hyperbolic trajectory to undergo an aerobraking maneuver at 100 km above Mars’s surface. The Mars reentry hyperbolic orbital trajectory has a semi-major axis a_{m1} = 2832.125302 km, an eccentricity e_{m1}=2.228759193, with Mar’s center as the hyperbola’s focus. It also has a specific angular momentum |ε_{t2}| = 7.561106136 km^{2}/sec^{2} and an eccentric anomaly F_{m1}=298.680726 Degrees. The angle between the asymptotes of the hyperbola Delt δ_{2} = 53.31806995 Degrees.

The Starship approaches the hyperbolic perigee r_{o2} = 3.380 x 10^{3} km. At the perigee, it has an angle between Mars’s orbital velocity vector and the perigee radius vector of η_{ 2} =93.749 degrees and velocity v_{o2}= 6.303650163 km/sec.

### Starship Approaches Mars

Again, The v_{o2} is less than the acceptable safe entry velocity indicated by SpaceX. So, the Starship has two options: Land or Park in Orbit. The decision is to park in orbit. So now, it heads towards Mars for an atmosphere reentry.

### Starship First Walk-in for Atmospheric Aerobraking

At the hyperbolic perigee, 100 km from Mars surface, the Starship experiences high drag caused by Mars atmosphere. This aerobraking caused by atmospheric drag shaves a considerable amount of its velocity. The ship exists the Martian atmosphere, with a velocity of v_{m2}=4.803650163 km/sec. It puts it on an elliptic orbit with Mars as its focus where the semi-major axis a_{m2}=2.783319446 x 10^{4} km, an eccentricity e_{m2}= 0.874969436, a specific angular momentum |ε_{m2}| = 0.769369108 km^{2}/sec^{2}.

The Starship needs 16.317 days to complete the orbit and be back where it started at the ellipse’s perigee. During this Orbit, SS will perform regular system checks and reference alignments and monitor Mars’s condition for atmospheric reentry.

### Starship Lands on Mars

Upon completing its orbit around Mars, the ships and Mars’s conditions are optimal for Landing. The Starship position itself and applies the atmospheric reentry maneuver. It starts the Mars Entry Maneuver Injection (MEMI), then does a free fall atmospheric entry. At around 15km altitude, the Starship positions itself into the Horizontal mode. Its raptor engines then ignite and initiate the entry burn, then proceeds with the aerodynamic guidance, and finally, land vertically on the surface of Mars.

## Last Words

I am constantly reminded that we are living in exciting times. SpaceX is working on landing a spaceship on the moon. The Starship is going through operation tests. And the whole space community and ecosystem is exponentially growing.

SpaceX has surprised us before, and I hope it will surprise us now and start by paving the way to Mars soon. I hope this article inspires people and allows them to build on it and improve it. This article is not the end; it is a start, and if you have constructive critique to improve what I did, please do not hesitate to contact me at o.alayli@futurespacearchitecture.com

Live long and prosper!

*Disclaimer: The Calculations in this article did not take into account Geocentric and Heliocentric gravitational degradation. Also, it did not take into account the impact of Venus’s gravitational field on the Starship. These issues will be addressed in future revisions.*