On the Requirements of a Generational Starship

In my first post relating to the idea of a multi-generational starship I outlined a mission plan.  Now we move beyond that initial plan to consider some of the mission requirements.  First and foremost must be consideration of the needs relating to long-term human habitation in deep space.  This problem has been investigated in the past.  Much of what follows is based on work done in the 1977 Princeton space colony survey: "Space Settlements: A Design Study."
The core difference in what follows is the population size.  The minimum population needed for a fully self sufficient colony able to support broad based manufacturing is approximately 500,000, with at least 200,000 to provide minimum commercial and agricultural services for a self-sufficient colony.  These numbers are discussed in the Princeton survey, however the colony size suggested there is a mere 10,000 individuals.  This earth orbiting colony acts as an in-situ construction platform for satellites and space vehicles.  Such a colony would be a necessary fore-runner to an interstellar generation ship, both as a construction facility and as an engineering and social test bed.
What follows is a more detailed analysis of those facilities necessary for a full sized generation ship.  Using this further size requirements on the final vehicle are developed.

Given a minimum population size of 500,000 and the expectation that the population will double in transit we have a design population of 1 million people.  This figure will be used throughout for all design.  Based on the Princeton survey the necessary floorspace in square meters per person for specific functions is:

• Residential: 49 sq.m/person
• Commercial/Community: 46 sq.m/person
• Agriculture: 60 sq.m/person

Further we can state that these functions have particular height requirements.  Residential areas require 3m of ceiling space, agricultural areas 15m.  Public areas height requirements vary depending on the particular function, ranging from 3-10m.
Rather than laying this entire plan flat it is a far better use of materials to build a vehicle with multiple "decks" accommodating some portion of the necessary facilities and the population.  Given the variable height of the public areas it also makes sense to build these decks to contain multiple floors, some areas being built to at most 3 floors, and some taking up more or less of the 10m space.
Given such a design for public areas it equally makes sense to split the residential areas similarly.  For the convenience of the populace we will intermingle residences and public areas.  These habitation decks are determined to be 12m in height.  Each such deck will be capable of accommodating up to 4 residential floors, with public spaces ranging from 3 - 12m in height with the assumed average height being 4m.  Parks, theaters, and similar areas will naturally be larger than schoolrooms and shops.  One might wonder about parks that area a scant 12m tall, it is therefore left to architectural and aesthetic concerns if perhaps such areas may extend beyond a single "deck" to stretch between multiple habitation decks, as they are connecting in a concentric fashion.  Likewise movement between decks is left to the imagination.  One can just as easily consider a "corkscrew" like layout, making the entire vessel a single extremely long corridor, as truly concentric decks connected only by lifts and ladders.
Given 4 residential floors and an average of 3 public area floors per deck the above floorspace requirements give a needed deck plan area of 27.5 sq.m/person (12.25 sq.m residential, 15.33 sq.m public).  Or, for 1 million people, a total of 27.5 million square meters of deck is needed.
Naturally for the comfort of the inhabitants the vehicle will be required to generate psuedo-gravity.  Lacking more exotic means, the vehicle will be designed as a cylinder which rotates about the ship's center axis.  It is desirable that the change in perceived gravity over the height of a floor or deck, called gravity gradient, be kept too low to be noticeable
The necessary square footage determined above can be used to determine an outer radius for the vessel.  The area, of course, is simply the circumference of the circle multiplied by the length of the vehicle.  Assume for the moment that deck height, and total habitation ring height, are much smaller than total radius.  This will, in addition to making an initial estimate easier, help reduce the gravity gradient.  Multiply this by the number of decks and one has an approximate floor area:

An infinite number of choices for radius, r, and length, l, will produce a number of decks, n, which satisfies the required deck plan area, A.  This decision can be, however, fairly arbitrary.  A simple spreadsheet can be used to arrive at an acceptable combination of terms.  An outer radius of 950m and length of 500m are selected, with 10 decks giving a comfortable 28 sq.m/person.
This, of course, is in addition to the requirement of some 900 million cubic meters of agricultural space not yet accounted for.  Simply using the remaining radius from the habitat ring gives 32 agricultural decks to provide the necessary 60 million square meters of agricultural space with each deck having a 15m ceiling.
This leaves another 350m of the total radius that is unused.  If 150m is used for storage and facilities that benefit from low gravity work, this gives a total of 130 million cubic meters of such facilities.  This also leaves the remaining 200m as a central area.  This can be left open to allow a 0-g construction area and landing or docking area for begin and end of mission activities.
A basic breakdown of this layout:

• 10 Habitation decks, 12m each: 120m, 422 million cu.m
• 32 Agriculture decks, 15m each: 480m, 900 million cu.m
• Low-g Facilities, 150m, 129.5 million cu.m
• 0-g Facilities, 200m, 62.8 million cu.m

Which produces the following totals:

• Internal Volume
• Without Center Area: 1,450 million (1.45 billion) cu.m
• With Center Area: 1,510 million (1.51 billion) cu.m
• Surface Area: 8.8 million sq.m

Later these totals, along with estimates regarding internal structure composition and material selection, will allow us to calculate the total mass of the populated portions of the final vehicle.