Space Rescue Mission

Launch worksheet

Launch solution 

The Scenario

You are on a desperate mission to save a crew that is stuck in orbit near Saturn. Your challenges include escaping from earth’s gravity, navigating to the planet Saturn, docking with the stray spacecraft and getting back to earth safely again with the rescued crew. Are you up for the job?

Math Scenario 1: Escaping from Earth’s Gravity

First you’ll need to escape Earth’s gravity and get into low orbit. Most of the mass of your rocket will be fuel, just to make it into orbit around earth.

There is a famous rocket equation called the Tsiolkovsky rocket equation that tells you how much mass is lost from a rocket to achieve a certain “delta-v” – that is, the change in speed required to get into low earth orbit or even escape earth’s gravity all together:

deltav = ve * ln(m0 / mf)

where deltav is the change in speed, ve is the exhaust velocity of the rocket, m0 is the rocket’s initial total mass including fuel, and mf is the rocket’s final mass without any fuel or propellant left.

We can also re-arrange this equation to find out the ratio between your initial and final rocket weight:

deltav = ve * ln(m0 / mf)

m0 / mf = exp(deltav / ve)

Let’s say you need a deltav of 9000 m/s to get into low earth orbit, and that your rocket’s exhaust velocity is 5000 m/s. We can work out the mass ratio:

m0 / mf = exp(deltav / ve)

m0 / mf = exp(9000 / 5000)

m0 / mf = 6.05

This means the initial rocket mass is more than 6 times that of the empty rocket once it has reached orbit. 

In your rescue mission, your rocket including engines, living quarters, rescue gear, supplies and general craft structure weighs 105 tonnes. To escape earth’s gravity completely we need a deltav of about 11200 m/s (ignoring air resistance). With the rocket specs as before, what will your total rocket mass be at launch?

m0 / mf = 6.05

m0 = 6.05 * mf

m0 = 6.05 * 105

m0 = 635.25 tonnes

Math Scenario 2: Gravity Assist

To get to Saturn, you’re going to need to use a few gravity assists. Gravity assists work by increasing your spacecraft’s velocity by approaching a planet and “slingshotting” around it, adding some of the planet’s orbital velocity to its own.

You’ve identified a few gravity assist maneuvers that will add an average of about 0.4 times your current velocity. If your current velocity in a sun reference frame is 20000 m/s, and you need an additional 15000 m/s of deltav to make it all the way to Saturn, how many gravity assists will you need?

First gravity assist:

First gravity assist end velocity = current velocity * 1.4

First gravity assist end velocity = 20000 * 1.4

First gravity assist end velocity = 28000 m/s

Deltav achieved so far = 8000 m/s

Second gravity assist:

Second gravity assist end velocity = current velocity * 1.4

Second gravity assist end velocity = 28000 * 1.4

Second gravity assist end velocity = 39200 m/s

Deltav achieved so far = 19200 m/s

Two gravity assists will be sufficient to achieve at least the required deltav of 15000 m/s.

Math Scenario 3: Aerobraking

When you get into the vicinity of Saturn, you’re travelling way too fast to dock with the stranded spacecraft. You don’t have enough spare fuel to use propulsion to slow down, so you’re limited to the highly experimental aerobraking

You figure you have about 50 orbits of the planet to bleed speed until you hit the required target speed of 8000 m/s.

If you approach the planet at a speed of 18000 m/s, what percentage of speed will you need to bleed off per orbit pass to get down to the target speed?

Final speed / initial speed = (percentage speed retained per orbit)^#orbits

log(Final speed / initial speed) = log[ (percentage speed retained per orbit)^#orbits]

log(Final speed / initial speed) = #orbits * log[ (percentage speed retained per orbit)]

log[ (percentage speed retained per orbit)] = log(Final speed / initial speed) / #orbits

percentage speed retained per orbit = 10^[ log(Final speed / initial speed) / #orbits ]

percentage speed retained per orbit = 10^[ log(8000 / 18000) / 50 ]

percentage speed retained per orbit = 0.9839

So you’ll need to bleed off about 1.61% of your speed per orbital pass.

Math Scenario 4: Radiation Exposure on the International Space Station

To launch your rescue mission, you’ve had to suspend missions to the International Space Station. The astronauts there are in danger of getting unhealthy amounts of radiation exposure.

The International Space Station orbits at an altitude of approximately 400 km.

Using the NASA graph below, 400 km is about 215 nautical miles, which gives a dosage of about 25 milliRad / day.

Without normal missions, the astronauts will be stranded there for 2 years:

Total dose = daily dose * time duration

Total dose = 25 milliRad/day * 2 years * 365 days / year

Total dose = 18.25 Rad

Radiation dosages at different altitudes. From NASA: https://www.nasa.gov/audience/foreducators/topnav/materials/listbytype/SMIII_Problem28.html

Real Life Example: Space Shuttle Columbia Disaster

Before the Space Shuttle program ended, it had flown a total of 135 missions, with two shuttles lost in mission accidents: Challenger in 1986 and Columbia in 2003.

The first Challenger disaster happened only seconds after launch and was catastrophic.

However, the second Columbia disaster occurred upon re-entry, even though the cause had happened days earlier when during launch a piece of foam insulation damaged the wing, weakening it for re-entry. 

If a rescue mission had been staged, it might have involved another shuttle being prepared to launch and dock with the stricken Columbia, while the crew on the Columbia went into a survival mode to minimize CO2 generation and other survival critical factors.

There were three launch windows – the times that would enable the earliest possible rendezvous with the Columbia.

Ultimately, such a rescue mission would have been at the best a very long shot, and as with the original Challenger disaster, there were apparently management issues at NASA that might have meant it would never have happened regardless.