The interstellar minimum

As a part of our educational initiatives, our team has recently put together a small test paper for starship design, known as 'the interstellar minimum' after the famous Russian physicist Lev Landau and his 'theoretical minimum' entrance exam for university degree programs. We are planning to produce various versions of this with different levels of difficulty, but so far no single person has been able to complete all of the questions correctly. So why not have a go and see if you really know about Starship design?

The Interstellar Minimum

Guidelines: You have been issued this test paper to challenge your knowledge of starship engineering. This specific paper is based around the Magellan Starship concept from the 1986 Arthur C Clarke novel “The Songs of Distant Earth”. You must email (info@i4is.org) your entry to us. Anyone who scores higher than 70% overall will win an I4IS prize and you will be hearing from us. You will also receive an invitation to join one of our unique educational or research committees. This is a test of your own knowledge and ability to solve these problems. Note that some of the problems follow on from the previous ones, in that answers to one question are input to another. You must answer all of the sections listed below and at least two questions from each section or your paper will not be accepted for marking. Please assume throughout that 1 ton = 1,000 kg, 1 AU = 1.49×10¹¹m. Good luck.

Section I: Fundamentals of Magellan

1. The Magellan travels to Thalassa 50 LY away at 0.2c, where 1LY=9.4605×10¹⁵ m and 1c=3×10⁸ m/s (i) Ignoring acceleration and deceleration, how long will it take the Starship to arrive at Thalassa in months (ii) if they transmit a radio signal back to Earth to say they have arrived, how long would the signal take to reach Earth in months?
2. Hypothetically assume that the Magellan has a total mass (structure+propellant) of 1,000,000 tons and it also has an additional payload mass of 1,000 tons. In order to achieve the 0.2c cruise velocity, what exhaust velocity must the engines be in the units of km/s if all of the structure mass is thrown away at destination end?
3. The Thrust (N) of the engines is given by multiplying by the mass flow rate (kg/s) and the exhaust velocity (m/s). (i) Assuming a mass flow rate of 0.5 kg/s, what is the Thrust of the Magellan engines? (ii) Assuming that kinetic energy = 0.5×m×v², when the starship approaches the target destination, prior to deceleration, and just assuming a payload mass alone, what will be the kinetic energy of the vehicle in Joules and Gtons TNT, where 1 kg TNT=4.18×10⁶ J?

Section II: Relavitistic Magellan

1. The Magellan starship took 250 years to get to Thalassa travelling at 0.2c.  From the perspective of observers on planet Earth, how long did the journey take?
2. A spacecraft travels away from Earth at speed 0.9c and fires a probe in same direction with speed 0.7c. What is probes speed relative to Earth?
3. If a scout ship leaves Earth at 0.95c and chases another ship which is moving at speed 0.9c, what is its speed relative to the spaceship?

Section III: Solar Sail Based Magellan

1. (i) Calculate the Solar irradiance (W/m²) at Earth orbit (1AU) and also very close to the Sun (0.01AU). (ii) Calculate the radiation pressure for the same distances assuming a 0.8 reflectivity.
2. Imagine that the payload mass of Magellan (1,000 tons) was deployed via a 1000,000,000 m² parabolic solar sail at 0.01 AU from the Sun. (i) What is the sail loading? (ii) the characteristic acceleration and lightness number.
3. Calculate (i) the escape velocity of the probe from the Sun (ii) how long it would take to reach Thalassa 50 light years away assuming that 1 LY = 9.4605×10¹⁵ m.

Section IV: Laser Sail Magellan

1. Calculate the beam spot size for a 1μm laser on a 200 km diameter lens system for distances of 10 AU, 100 AU, 1,000 AU and 50 LY to Thalassa.
2. Assuming an acceleration of 1g=9.81 m/s², a sail reflectivity of 0.8 for a 1,000 tons spacecraft, what is the power required to push the sail?
3. Assuming a sail loading of 0.01 kg/m2 for our 1,000 tons probe, what is the diameter of a circular sail required for the mission?

Section V: Nuclear Propulsion Magellan

1. Using the derived earlier exhaust velocity for Magellan of 8,685 km/s and the assumed mass flow rate of 0.5 kg/s and calculated thrust of 4.34 MN, calculate the Jet power of the vehicle.
2. Now calculate the specific power for Magellan assuming that the 1,000 tons payload reaches the target and the propulsion mass is 100 tons.
3. If the Magellan specific power was actually 1 MW/kg for a 1,000 tons probe with 100 tons propulsion mass, and assuming the same thrust of 4.34 MN (i) what would be the new exhaust velocity? (ii) assuming the same mass ratio term of Ln(R) = 6.908 derived earlier and the revised exhaust velocity, calculate the new cruise velocity of the vehicle and estimate how long it would take to get to Thalassa.

The Initiative for Interstellar Studies
Educational Academy Committee