Premise: A generation spaceship leaves Earth around the year 2060 on a journey to colonize Alpha Centauri A (ACA). In this fiction, fusion power is achieved in 2040, improved over 20 years, and used within the solar system. The trip to ACA will take 110 years. The ship will accelerate halfway, flip, and decelerate for the second half.

I understand basic physics equations involving $F (force) = m (mass) * a (acceleration)$ and simplified space travel using constant acceleration giving $d=(1/2)at^2$, with distance (d) in meters, acceleration (a) in meters per second squared, and time (t) in seconds.

However, this distance traveled does not account for mass loss of Xenon fuel used for propulsion. How do I set up an equation to get (at least a rough estimate of) the Newtons of thrust and kg of Xenon needed for the journey to take 110 years?

Given:

• The ship leaves in 2060: about 40 years more advanced than our current 2021 tech levels.
• The journey takes 110 years (as relatively perceived by those on board the ship).
• Ship launch mass of 1,900,000 kg.
• Each ion drive provides 30 N thrust, averaging 15 kW used per N, fuel use 75 kg of Xenon per 4,000 seconds of burn. (based on advanced versions of current drives)
• Light years to ACA: 4.37.

Edit: thanks to answers and comments : Originally, I thought they would flip the ship to decelerate halfway, but the ship will want to continue to burn at the same max safe thrust, and so burn near constant fuel during the entire trip. So, the latter half of the trip will see increasingly larger acceleration, due to decreasing mass but constant thrust, Newtons. This changing mass makes the calculation more complex, because they will not simply flip at halfway point... as the deceleration part will be shorter due to lower mass. I am currently researching rocket equations which account for fuel mass losses but don't have it figured out yet...

Journey with simplified acceleration if time is 110 years: $a = d/0.5t^2 = (2.06717e16) / (0.5 * (3.469e9)^2) = 0.00343556041 m/s^2 = a$.

If the ship is 1,900,000 kg at launch from Earth, and $F=ma$, $1900000*a = 6527$ N (Newtons of thrust). However this is simplified. N thrust will change as fuel mass is lost... My thinking is that the ship will want to continue to burn at the same max safe thrust, and so burn near constant fuel during the entire trip. So the latter half of the trip will see increasingly larger acceleration, due to decreasing mass but constant thrust.

6527N can be provided by 218 individual 30N drives (around this number may be good even as mass lessens, for redundancy safety). Based on above givens, this requires 861,110 kg Xe fuel. Ship mass would continually decrease as Xe used, until the ship is empty of fuel and about 1,040,000 kg mass remains, requiring less force to move.

I'm not sure how to estimate how much N of thrust and mass of Xe fuel will be needed for this journey. I am imagining two functions, with the force function relying on the lost Xe mass (which is a constant loss over time), but I am unsure how to set that up so that everything results in a 110 year journey. Should I integrate to get areas underneath both functions, then adjust until I get roughly 110 years? Ideally I'd like equations where I can easily adjust the ship mass, thrust Newtons, and so on to calculate with different variables if needed.

I am also disregarding initial velocity in the above equations. Ideally for the story, the ship would leave from Mars orbit, and here are some more givens: Linear distance can be expressed as (if acceleration is constant): $s = v_0t + 1/2 a t^2$

• $v_0$ = initial linear velocity (m/s) = Mars mean orbital velocity in (m/s) = $24070$

Some things I researched include:

Info and chart below from https://en.wikipedia.org/wiki/Ion_thruster#Comparisons

Ion thrusters in operational use typically consume 1–7 kW of power, have exhaust velocities around 20–50 km/s (Isp 2000–5000 s), and possess thrusts of 25–250 mN and a propulsive efficiency 65–80%.[3][4] though experimental versions have achieved 100 kW (130 hp), 5 N (1.1 lbf).[5]

https://solarsystem.nasa.gov/missions/dawn/technology/spacecraft/ Dawn Ion Propulsion System Number of thrusters: 3 Thruster dimensions (each): 13 inches (33 centimeters) long, 16 inches (41 centimeters) in diameter Weight: 20 pounds (8.9 kilograms) each Spacecraft acceleration via ion propulsion at full thrust: 0 – 60 mph in 4 days Thrust: 0.07 to 0.33 ounce (19 to 91 millinewtons)

Fuel https://en.wikipedia.org/wiki/Ion_thruster#Propellants Many current designs use xenon gas, as it is easy to ionize, has a reasonably high atomic number, is inert and causes low erosion. However, xenon is globally in short supply and expensive. VASIMR design (and other plasma-based engines) are theoretically able to use practically any material for propellant. However, in current tests the most practical propellant is argon, which is relatively abundant and inexpensive.

https://en.wikipedia.org/wiki/Variable_Specific_Impulse_Magnetoplasma_Rocket [Higher energy use ok because of fusion power.] Other propellants, such as bismuth and iodine, show promise, particularly for gridless designs such as Hall effect thrusters. Krypton is used to fuel the Hall effect thrusters aboard Starlink internet satellites, in part due to its lower cost than conventional xenon propellant. FUEL USE: The Deep Space 1 spacecraft, powered by an ion thruster, changed velocity by 4.3 km/s (2.7 mi/s) while consuming less than 74 kg (163 lb) of xenon. [=4300 m/s for 75kg Xe?] The Dawn spacecraft broke the record, with a velocity change of 11.5 km/s (41,000 km/h), though it was only half as efficient, requiring 425 kg (937 lb) of xenon.

https://www.space.com/38444-mars-thruster-design-breaks-records.html https://www.popularmechanics.com/space/moon-mars/news/a28754/new-ion-thruster-breaks-records-power-thrust/ https://www.space.com/28732-nasa-dawn-spacecraft-ion-propulsion.html https://www.nasa.gov/centers/glenn/technology/Ion_Propulsion1.html https://www.nasa.gov/multimedia/imagegallery/image_feature_2416.html How fast will 1g get you there? http://www.projectrho.com/public_html/rocket/slowerlight2.php http://www.xenology.info/Xeno/17.3.htm Conventional Interstellar Propulsion Systems https://forum.nasaspaceflight.com/index.php?topic=34036.1060 https://www.omnicalculator.com/physics "The Martian" Hermes ship design https://the-martian.fandom.com/wiki/Hermes_Spacecraft https://www.nasa.gov/directorates/spacetech/niac/index.html