SURVIVAL
SKILLS
WHAT TO DO WHEN YOUR INTERSTELLAR CRUISE GOES WRONG !
(An earlier version of this presentation was given at
the 2004 Summer Lecture at the Perth Observatory)
HYPOTHETICAL - OR A SHORT LESSON IN ASTROPHYSICS
How far away is the Sun?
How do we know how far away the Sun is? After all, the Moon and the Sun look almost identical in size to us from the surface of the Earth. We can tackle the problem in a different way. Imagine that we are interstellar travellers whose spacecraft suffers a tragic fate in warp space (whatever that may be). We make it into a lifecapsule, and eventually we land on an unknown planet in an unchartered part of the galaxy. Fortunately we find the atmosphere breathable, and there appears to be an abundance of edible fruits and vegetables.
After having overcome the shock of being stranded in a most alien land, our scientific curiosity sets us to explore the system in which we have arrived. We see two bodies in the sky. One is obviously a sun - we know this immediately from the intensity of the light and heat emanating from it. The other smaller body is equally obviously a moon which shines by the reflected light of the sun.
Having no radar, we cannot measure distances. we can however, construct a crude theodolite which will enable us to measure angles, and our galactic watch, which did survive, will allow us to measure times.
The first thing we need to do is to measure a rotation rate for our planet. We do this by observing the transit of one or more bright stars at night. What we find is that the star alpha-3 (there is a pattern of stars in this sky that look like the number three, and this is the brightest star of the group) passes the same mark we have set up in 80,000 standard seconds (a little over 22 Earth hours). This gives us the planet's sidereal rotation rate. We then time a solar transit on two successive days and find that the synodic rotation rate (with respect to our sun) is 80,268 seconds. We realise that this second value is only approximate because it is unlikely that the planet is in an exactly circular orbit around the sun, but rather an ellipse. This means that the solar transit time will vary throughout the year, as sometimes the sun will appear to be fast, and at other times slow with respect to a mean solar day.
The difference between the sidereal and synodic rotation rates will allow us to make an estimate of the length of the planet's year. The reason for the difference between the two rates is because the planet is moving around the sun at the same time that is rotating about its axis. This means that for the sun to appear in the same place the next day, the planet must rotate a little bit more than 360 degrees (see slide below). The extra rotation is 268/80000*360 degrees. This comes out to 1.21 degrees. Now in a year the planet must rotate a full extra 360 degrees to come back to the same place and orientation that it had the year before. It will take 360/1.21 or 298 earth days to do this. So our year is 298 days long.
Knowing the sidereal rate, we can perform a transit observation of both the sun and the moon and with a bit of geometry (to account for the fact that they are not passing us overhead) we can determine a rough value for their respective angular diameters. The sun is 2310 seconds of arc, and the moon is 1000". When we know the distance to these bodies we will be able to quickly calculate their diameters.
The acceleration due to gravity on the surface of the planet can be measured very roughly with spring scales or more accurately by timing the period of a pendulum (P).
so g = 4 π2 L / P2
Once g has been determined, we can calculate the ratio between the planet's mass and its radius squared:
where G = 6.67x10-11 is the universal constant of gravitation. What we then need is the radius R of the planet. This can be very roughly determined by noting the horizon distance. Once we have R we can use the above equation to find M. Knowing M we can then use Newton's laws to find the distance to the moon which revolves around the planet (this depends only on the mass of the planet - not the moon - and its distance from the planet).
We can then use a little geometry to find the distance to the sun. Knowing this, and the length of the planetary year, we can find the mass of the sun. The angular diameter of the sun then gives us its true diameter.
We might put a black tray filled with water in the direct sunlight and obtain an estimate of the amount of radiation that the planet receives from the sun. Knowing this it is simple to compute the temperature of the sun's surface.
A simple physical model of the star's energy source using the ideal gas equation can even give us a not too bad estimate of the sun's core temperature.
And so we proceed. Measurements are crucial in all this, particularly measurements of distance. The more precisely we can measure angles and time, the more accurately we can find distances.
NOW TO THE STORY - LOST IN SPACE
Unfortunately for the hero of our story he is the only person who makes it into the space life boat and following the instructions he finds a close star, and is even able to see a planet around the star to which he then guides the life boat down to its surface for a successful landing. Unfortunately all other souls were lost in space.
After spending some time reading the life boat manual, he manages to figure out how to test the atmosphere with some of the onboard equipment. It appears that he will be able to survive on the planet without having to wear a spacesuit. What a relief that is!
He gingerly opens the life boat hatch when the outside pressure indicator shows a value that is within 10% of the interior pressure. He steps out of the life boat to a world that doesn't smell too bad, is at a pleasant temperature , and has a 'sun' in the sky that is only a little redder than the Earth's Sun.
He spots what look like palm trees in the distance and even some fruit hanging from beneath their fronds. Walking towards them he stubs his foot on something that on inspection appears like an edible tuber! What luck!. He wonders if the natives are peaceful but can see no sign of any living creature.
After resting for a few days and exploring in the vicinity of the life boat, he decides that he really needs to do something to find where in the galaxy he is, and identify the star around which the planet revolves. To do this he realises (with some help from the survival manual all good Galactic Cruiser life boats carry) that the first thing he needs to do is take some transit measurements.
Erecting two vertical poles he uses the life boat chronometer to measure the number of Earth seconds in the planet's day. Not only the 'solar' day but also the sidereal (or star) day. The latter gives him the rotation period of the planet (Ts). This is the time that it takes the planet to rotate through 360 degrees. So he starts the chronometer stop watch function on the first night a bright star lines up with the two poles he erected and stops it on the exact same point the following night. The moon provides some light to help with this. He then does the same thing for the 'sun' in the day sky.
Because the planet is revolving around the "sun", the duration of a solar day is a little longer than a sidereal or star day. This is because the planet has to rotate a little bit more than 360 degrees to bring it back into line with the sun on the second day - because at this time the planet has moved along its orbital path (the next figure should help you see this).
Out intrepid traveller is now able, from his two measurements, find the length of the day and also the year. The next thing to do is to find the mass of the planet. To do this he first needs to find the gravity on the planet's surface.
The acceleration due to gravity is then given by the formula:
The next step is to find the radius of the planet. To do that we need to locate a reasonable sized sea on the planet. This will give a sea level as a base. The survivor is lucky enough to not only find a sea, but also finds a palm tree growing on the beach. This will allow him to determine the distance of the horizon from a high hill.
Walking back from the beach toward such a hill he climbs the hill until the base of the palm tree in the distance just disappears from view as seen in his Galactic binoculars.
Even though he is very tired from all the walking he uses the formula in the survival manual to compute the radius of the planet. He is then also able to calculate the mass of the planet from the formula:
On a roll he is then able to calculate the distance to the planet's moon by Newton's second law:
where ω is the angular rotation of the moon about the planet. This can be timed with allowance made for the planetary rotation.
The next step is to find the distance to the sun. This is the most difficult measurement to make because the angle is so close to and just under 90 degrees.
It cannot be an angular measurement because when the phase of the moon is such that it is exactly half illuminated the sun will not be in the sky. So it has to be a timing measurement - with due allowance made for the movement of the sun due to the earth's rotation and revolution.
Knowing the distance to the 'sun' and its angular size (from a transit measurement) he can easily calculate the diameter of the 'sun'.
The last measurement our lost traveller needs to make is the amount of energy coming from the 'sun'. A simple but rather inaccurate way to do this is to time how long it takes for the temperature of water in a blackened tray to rise a certain amount. It is inaccurate because it is difficult to account for all the heat losses involved. It also requires a thermometer.
Let us assume that we surmount all these difficulties. Now if the energy flux from the 'sun' is written as S (watts per square metre), the amount of heat energy absorbed by the water in the tray will be Hin = S A cos θ where A is the surface area of the tray and the 'cos θ' term is due to the fact that the 'sun' is not overhead. The units of Hin are watts.
It would be best if we performed this experiment around noon so that the sun angle varies as little as possible. We must also realise that while the temperature of the water is equal to the ambient temperature Tamb there will be no heat loss to the environment, but as soon as the water temperature T rises above Tamb heat will start to flow out of the water into the environment. We will call this the heat loss Hloss and this will also be measured in watts.
The difference between the heat in from the 'sun' and the heat loss is what causes the water to warm up, according to the formula:
Now can rewrite our equation to find S:
The problem with this formula is that we don't know anything about the heat loss term (Hloss) - EXCEPT at time zero when it is also zero.
So what we need to do is to take measurements say every minute and plot a graph of temperature difference ( ΔT = T - Tamb) versus time from the start (Δt = t - t0). This will start to rise rapidly at first and then slow down and eventually become a horizontal line when the heat loss equals the heat input. We don't need to plot for that long however, because we are only interested in the slope of the curve at the start.
 | Plot of water temperature increase with time. At the start, when heat losses are zero, the temperature rises at its fastest rate. At the end the temperature reaches an equilibrium when the heat loss from the tray equals the heat input from the 'sun'. The red line is the tangent to the curve at the start, and is the value we use to give us the best estimate of the heat flux from the 'sun'. |
So we need to draw a straight line from the origin of the curve (T=0, t=0) and draw the line tangential to the curve at the start. The slope of this line is [ΔT/Δt]t=0. When we do this we can immediately calculate S:
The inaccuracy in this is that we assume all the sun's heat will be absorbed by the water, when in practice the water will reflect some of the heat back into the sky.
If you think this is all too difficult it might be a lot better, easier and more accurate to use the stellar photometer which is standard issue on all Galactic life boats, and which gives the flux of the 'sun' directly in watts per square metre.
Knowing the solar flux at the planet we can calculate the total energy output of the Sun from
This is in watts. Out last formula will allow us to calculate the surface temperature of the 'sun':
so
So our stranded traveller now has all the information he needs about his planet and 'sun'.
With all these measurements the traveller is now in a position to use this information to find the possible candidate stars from the Galactic Stellar Catalog carried by all space life boats. Most importantly, he can point his subspace radio beacon toward the Earth.
REFERENCE
Australian Space Academy