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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...

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.

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]

Edit: thanks to answers and comments : Originally, I thought they would flip the ship to decel 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 accel, 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 decel part will be shorter due to lower mass. I am currently researching rocket equations which account for fuel mass losses but dont have it figured out yet...

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 accel, due to decreasing mass but constant thrust.

Regarding initial velocity: Ideally for the story, the ship would leave from Mars orbit: Linear distance can be expressed as (if acceleration is constant): $s = v_0 * t + 0.5a t^2$. With $v_0 =$ initial linear velocity (m/s) = Mars mean orbital velocity in (m/s) = $24070$

Regarding relative movement of both the Solar System and Alpha Centauri, I found:

Using spectroscopy the mean radial velocity has been determined to be around 22.4 km/s towards the Solar System. This gives a speed with respect to the sun of 32.4 km/s, very close to the peak in the distribution of speeds of nearby stars.

But without knowing ship's max v, because the ship-flipping point is unknown to me, I'm not sure how much 22.4 kps will affect the journey.

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]

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

However, this distance travelled 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?

Givens:

• 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 30N thrust, averaging 15kW used per N, fuel use 75kg of Xenon per 4,000 seconds of burn. (based on advanced versions of current drives)
• Lightyears to ACA: 4.37.

Edit: thanks to answers and comments : Originally, I thought they would flip the ship to decel 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 accel, 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 decel part will be shorter due to lower mass. I am currently researching rocket equations which account for fuel mass losses but dont have it figured out yet...

Journey with simplified acceleration if time is 110 years: $a = d/.5t^2 = (2.06717e16) / (.5 * (3.469e9)^2) = 0.00343556041 m/s^2 = a$.

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 accel, due to decreasing mass but constant thrust.

I'm not sure how to estimate how much N of thrust and kg 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, thruster 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): $s = v0 * t + 1/2 a t^2$

• v0 = initial linear velocity (m/s) = Mars mean orbital velocity in (m/s) = $24070$

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.

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) = $24070$

• 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 30N thrust, averaging 15kW used per N, fuel use 75kg of Xenon per 4,000 seconds of burn. (based on advanced versions of current drives)
• Lightyears to ACA: 4.37. Half the journey until flipping to decel = $2.185 Lyrs = 2.06717e16$ meters.

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 required will change as fuel mass is lost...

• 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 30N thrust, averaging 15kW used per N, fuel use 75kg of Xenon per 4,000 seconds of burn. (based on advanced versions of current drives)
• Lightyears to ACA: 4.37.

Edit: thanks to answers and comments : Originally, I thought they would flip the ship to decel 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 accel, 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 decel part will be shorter due to lower mass. I am currently researching rocket equations which account for fuel mass losses but dont have it figured out yet...

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 accel, due to decreasing mass but constant thrust.