Apollo 11's Translunar Trajectory
and how they avoided the radiation belts

Robert A. Braeunig
© November-2009


Most attempts to illustrate the trajectories of the Apollo flights to the Moon are intended to show only the primary elements of the mission, with little attention to geometric accuracy and scale. One such example is shown below. Unlike these common not-to-scale drawings, this page takes a new approach. After determining the orbital elements of Apollo 11's translunar trajectory, I've calculated its position versus time and accurately plotted, in correct proportion and orientation, its flight path to the Moon. Furthermore, Apollo 11's trajectory through the region of the Van Allen Radiation Belts has been mapped to show how the trajectory was designed to bypass the most intense areas of the this potentially dangerous obstacle.

Orbital Elements

If you don't like math, I recommend you skip ahead because this next part is going to be math intensive. I've included the calculations for those who might be interested in checking my work and/or learning the computational methods.

All data used in the calculations comes from the following:

Apollo by the Numbers - Translunar Injection

And, unless noted otherwise, all mathematical equations and procedures come from another of my web pages:

Rocket & Space Technology - Orbital Mechanics

From the Apollo by the Numbers web page, we have the following data for Apollo 11 at the time of Translunar Injection (TLI):

GET: 002:50:13.03
KSC Date: 16-Jul-1969
GMT Date: 16-Jul-1969
KSC Time: 12:22:13 PM
Time Zone: EDT
GMT Time: 16:22:13

Altitude (ft): 1,097,229
Altitude (n mi): 180.581
Earth-Fixed Velocity (ft/sec): 34,195.6
Space-Fixed Velocity (ft/sec): 35,545.6
Geocentric Latitude (deg N): 9.9204
Geodetic Latitude (deg N): 9.983
Longitude (deg E): -164.8373

Flight Path Angle (deg): 7.367
Heading Angle (deg E of N): 60.073
Inclination (deg): 31.383
Descending Node (deg): 121.847
Eccentricity: 0.97696
C3 (ft2/sec2): -14,979,133

Although this is a good start, we need to do more to obtain the six elements needed to fully describe the orbit. Referring to my Orbital Mechanics web page, let's use the method described in the section titled "Launch of a Space Vehicle". The flight path angle, f, and velocity, v are both given, which are 7.367o and 35,545.6 ft/s respectively. Let's first convert velocity to SI units (mks):

v = 35,545.6 × 0.3048 = 10,834.3 m/s

From the altitude we can calculate the length of the radius vector, knowing that altitude is the perpendicular distance from the vehicle center of gravity to the Fischer Ellipsoid of 1960. The radius of the ellipsoid is given by the formula,

R = a × b / SQRT[ (b × cos)2 + (a × sin)2 ]

Where a is the equatorial radius of Earth, b is the polar radius of Earth, and is the geocentric latitude. The accepted values of a and b in 1960 were 6,378,166 m and 6,356,784.3 meters, respectively. The ellipsoid radius at the given latitude of 9.9204o is,

R = 6378166 × 6356784.3 / SQRT[ (6356784.3 × cos(9.9204))2 + (6378166 × sin(9.9204))2 ] = 6,377,528 m

Therefore, the magnitude of Apollo 11's radius vector at TLI was (converting nautical miles to meters),

r = 6,377,528 + (180.581 × 1,852) = 6,711,964 m

From equations (4.29), (4.30) and (4.31) we calculate the perigee and apogee radii, the eccentricity, and the true anomaly:

C = 2 × 3.986005E+14 / (6,711,964 × 10,834.32) = 1.011851

(Rp/r) = (-C + SQRT[ C2 - 4 × (1 - C) × -cos2(7.367) ] ) / (2 × (1 - C)) = 0.9833646
Rp = 0.9833646 × 6,711,964 = 6,600,308 m

(Ra/r) = (-C - SQRT[ C2 - 4 × (1 - C) × -cos2(7.367) ] ) / (2 × (1 - C)) = 84.39725
Ra = 84.39725 × 6,711,964 = 566,471,300 m

e = SQRT[ (6,711,964 × 10,834.32 / 3.986005E+14 - 1)2 × cos2(7.367) + sin2(7.367) ] = 0.976965

n = arctan[ (6,711,964 × 10,834.32 / 3.986005E+14) × cos(7.367) × sin(7.367) / ( (6,711,964 × 10,834.32 / 3.986005E+14) × cos2(7.367) - 1 ) ] = 14.9087o

The semi-major axis is simply the average of the perigee and apogee radii:

a = ( 6,600,308 + 566,471,300 ) / 2 = 286,535,800 m

Next, we need a method to transform coordinates from one plane to another, which we accomplish using the following (from Astronomical Formulae for Calculators, 4th edition, by Jean Meeus):

tan a = ( sin l × cos i - tan b × sin i ) / cos l

sin d = sin b × cos i + cos b × sin i × sin l

where a = equatorial longitude, d = equatorial latitude, l = orbital longitude, b = orbital latitude, and i = inclination of the orbital plane. b is, of course, always equal to zero because the spacecraft never leaves the orbital plane. Setting b = 0, the equations simplify to the following:

tan a = tan l × cos i

sin d = sin l × sin i

These transformations use the location of the ascending node as the origin for measuring longitude. Once the transformation is complete, the resulting answer must be converted to either geographic longitude or celestial longitude, as required.

We're given the inclination and the geocentric latitude of the TLI launch point, which are 31.383o and 9.9204o N respectively. From this we calculate l and a as follows:

l = arcsin[ sin(9.9204) / sin(31.383) ] = 19.3189o

a = arctan[ tan(19.3189) × cos(31.383) ] = 16.6614o

The argument of perigee, w, is equal to l of the launch point minus the true anomaly:

w = 19.3189 - 14.9087 = 4.4102o

The geographic longitude of the ascending node equals the geographic longitude of the launch point, -164.8373o E, minus the launch point a:

Geographic longitude of ascending node = -164.8373 - 16.6614 + 360 = 178.5013o E

The celestial longitude of the ascending node, W, is equal to the local sidereal time at the geographic longitude at the time of TLI:

W = LST @ 178.5013o E, 16-Jul-69, 16:22:13 GMT = 23h 53' 31" = 358.380o

Apollo by the Numbers gives the longitude of the descending node as 121.847o, though exactly what this number represents is not explained. After a considerable amount of investigation and number crunching, I've concluded this is the degrees of longitude east of the launch site at GET=0. Since the sidereal time over Launch Pad 39A at GET=0 was 56.604o, the celestial longitude of the descending node is 56.604 + 121.847 = 178.451. The ascending node is 180o opposed at 358.451o. This checks very closely with my calculated value of 358.380o. The reason for the difference is unknown; however, I'm using the calculated valve to maintain internal consistency between all the numbers.

The only orbital element remaining is the time of perigee passage. Referring to the Orbital Mechanics page, we'll use the method described in the section titled "Position in an Elliptical Orbit". First, we calculate the time between the last perigee and the spacecraft's true anomaly at TLI:

E = arccos[ (0.976965 + cos(14.9087)) / (1 + 0.976965 × cos(14.9087)) ] = 0.0282451 radians

M = 0.0282451 - 0.976965 × sin(0.0282451) = 0.000654295 radians

n = SQRT[ 3.986005E+14 / 286,535,8003 ] = 0.00000411624 rad/s

t = 0.000654295 / 0.00000411624 = 159 s

The time of perigee passage is, therefore, the time of TLI minus 159 seconds:

T = 002:47:34 GET = 16:19:34 GMT 16-Jul-69

Recapping, Apollo 11's translunar orbital elements were as follows:

Locating Apollo 11

Now that we have the orbital elements, we can produce the calculations needed to locate Apollo 11 in three-dimensions. In general, the procedure is: (1) select a time, (2) calculate the true anomaly (by first calculating the mean anomaly and eccentric anomaly), (3) calculate the radial distance, (4) calculate the longitude in the orbital plane, (5) transform to equatorial longitude, (6) convert to celestial longitude, and (7) calculate the equatorial latitude. And, for information, let's also (8) calculate the flight path angle and (9) calculate the space-fixed velocity. For example, suppose we want to perform the calculations for 004:00:00 GET; we go about it like this:

Time since perigee passage = 4 × 3600 - (2 × 3600 + 47 × 60 + 34) = 4,346 s

Mean anomaly, M = 4,346 × 0.00000411624 = 0.0178892 radians

Eccentric anomaly, E = 0.0178892 + 0.976965 × sin E
by iteration, E = 0.382848 radians

True anomaly, n = arccos[ (cos(0.382848) - 0.976965) / (1 - 0.976965 × cos(0.382848)) ] = 121.765o

Radius, r = 286,535,800 × (1 - 0.9769652) / (1 + 0.976965 × cos(121.765)) = 26,866,200 m

Flight path angle, f = arctan[ 0.976965 × sin(121.765) / (1 + 0.976965 × cos(121.765)) ] = 59.684o

Velocity, v = SQRT[ 3.986005e+14 × 286,535,800 × (1 - 0.9769652) ] / (26,866,200 × cos(59.684)) = 5,318 m/s

Orbit longitude, l (from ascending node) = 121.765 + 4.4102 = 126.175o

Equatorial longitude, a (from ascending node) = arctan[ tan(126.175) × cos(31.383) ] + 180* = 130.581o

Equatorial longitude, a (celestial) = 358.380 + 130.581 - 360 = 128.961o

Equatorial latitude, d (celestial) = arcsin[ sin(126.175) × sin(31.383) ] = 24.858o

(* The arctan function is problematic because calculators and computers cannot distinguish the quadrant of the angle. For instance, arctan(1) = 45o and 225o. Care must be taken when working with coordinates to make certain the angle is placed in the correct quadrant.)

Using the procedure outlined above, the following table was produced. The polar coordinates of distance, longitude and latitude can be used to pinpoint Apollo 11's position in space at any time. It should be noted that all the calculations in this web page treat the orbit as a simple two-body program, i.e. Earth and the spacecraft. Apollo 11 is assumed to coast freely in space under the gravitational influence of only Earth. The gravity of the Moon, and all other perturbing forces, are ignored.


Apollo 11 - Translunar Phase
GET
(h:m:s)
Greenwich Mean Time
(date, time)
True Anomaly
(degrees)
Flight Path
Angle (deg)
Velocity
(m/s)
Distance from
Earth (m)
Equatorial Coordinates (Celestial)
Longitude (o) Latitude (o)
002:50:13 7/16/69, 16:22:13 14.9087 7.367 10,834.3 6,711,964 15.0416 9.9204
003:00:00 7/16/69, 16:32:00 58.5356 28.894 9,532.0 8,641,760 57.4909 27.6309
003:15:00 7/16/69, 16:47:00 91.1895 44.913 7,646.2 13,318,712 94.9318 31.2164
003:30:00 7/16/69, 17:02:00 106.5866 52.398 6,531.9 18,095,084 112.5874 29.0897
003:45:00 7/16/69, 17:17:00 115.6544 56.766 5,819.2 22,613,282 122.5194 26.7882
004:00:00 7/16/69, 17:32:00 121.7654 59.684 5,318.1 26,866,368 128.9617 24.8574
004:15:00 7/16/69, 17:47:00 126.2411 61.804 4,941.4 30,889,122 133.5460 23.2717
004:30:00 7/16/69, 18:02:00 129.7060 63.431 4,644.6 34,715,890 137.0177 21.9543
004:45:00 7/16/69, 18:17:00 132.4955 64.731 4,402.6 38,374,976 139.7650 20.8413
005:00:00 7/16/69, 18:32:00 134.8077 65.800 4,200.0 41,888,995 142.0111 19.8860
005:15:00 7/16/69, 18:47:00 136.7677 66.700 4,027.0 45,276,043 143.8935 19.0543
005:30:00 7/16/69, 19:02:00 138.4588 67.470 3,876.7 48,550,764 145.5025 18.3215
005:45:00 7/16/69, 19:17:00 139.9392 68.139 3,744.5 51,725,157 146.8997 17.6690
006:00:00 7/16/69, 19:32:00 141.2507 68.728 3,626.8 54,809,187 148.1289 17.0828
006:30:00 7/16/69, 20:02:00 143.4830 69.719 3,425.5 60,738,374 150.2036 16.0682
007:00:00 7/16/69, 20:32:00 145.3262 70.526 3,258.3 66,391,605 151.9006 15.2155
007:30:00 7/16/69, 21:02:00 146.8855 71.198 3,116.2 71,808,685 153.3254 14.4842
008:00:00 7/16/69, 21:32:00 148.2297 71.770 2,993.2 77,020,296 154.5461 13.8470
008:30:00 7/16/69, 22:02:00 149.4063 72.264 2,885.3 82,050,701 155.6090 13.2843
009:00:00 7/16/69, 22:32:00 150.4489 72.695 2,789.4 86,919,495 156.5468 12.7819
009:30:00 7/16/69, 23:02:00 151.3824 73.076 2,703.3 91,642,791 157.3832 12.3294
010:00:00 7/16/69, 23:32:00 152.2255 73.415 2,625.4 96,234,035 158.1363 11.9184
010:30:00 7/17/69, 00:02:00 152.9929 73.719 2,554.4 100,704,600 158.8196 11.5427
011:00:00 7/17/69, 00:32:00 153.6957 73.993 2,489.3 105,064,209 159.4439 11.1971
011:30:00 7/17/69, 01:02:00 154.3432 74.242 2,429.2 109,321,264 160.0177 10.8777
012:00:00 7/17/69, 01:32:00 154.9426 74.470 2,373.6 113,483,080 160.5478 10.5810
013:00:00 7/17/69, 02:32:00 156.0206 74.870 2,273.3 121,545,956 161.4986 10.0452
014:00:00 7/17/69, 03:32:00 156.9668 75.211 2,185.1 129,295,990 162.3305 9.5726
015:00:00 7/17/69, 04:32:00 157.8077 75.504 2,106.6 136,767,677 163.0679 9.1510
016:00:00 7/17/69, 05:32:00 158.5629 75.759 2,036.0 143,989,147 163.7286 8.7711
017:00:00 7/17/69, 06:32:00 159.2470 75.983 1,972.0 150,983,728 164.3260 8.4260
018:00:00 7/17/69, 07:32:00 159.8713 76.180 1,913.6 157,771,041 164.8703 8.1102
019:00:00 7/17/69, 08:32:00 160.4448 76.354 1,859.8 164,367,793 165.3695 7.8195
020:00:00 7/17/69, 09:32:00 160.9747 76.510 1,810.2 170,788,358 165.8300 7.5505
021:00:00 7/17/69, 10:32:00 161.4666 76.649 1,764.0 177,045,222 166.2571 7.3002
022:00:00 7/17/69, 11:32:00 161.9253 76.773 1,720.9 183,149,320 166.6549 7.0665
023:00:00 7/17/69, 12:32:00 162.3547 76.884 1,680.6 189,110,296 167.0270 6.8474
024:00:00 7/17/69, 13:32:00 162.7582 76.984 1,642.7 194,936,715 167.3762 6.6413
026:00:00 7/17/69, 15:32:00 163.4980 77.155 1,573.1 206,215,699 168.0159 6.2627
028:00:00 7/17/69, 17:32:00 164.1625 77.293 1,510.6 217,038,742 168.5896 5.9220
030:00:00 7/17/69, 19:32:00 164.7650 77.403 1,453.9 227,448,909 169.1092 5.6126
032:00:00 7/17/69, 21:32:00 165.3156 77.491 1,402.1 237,482,107 169.5836 5.3294
034:00:00 7/17/69, 23:3200 165.8224 77.559 1,354.3 247,168,686 170.0198 5.0684
036:00:00 7/18/69, 01:32:00 166.2915 77.609 1,310.1 256,534,595 170.4233 4.8266
038:00:00 7/18/69, 03:32:00 166.7280 77.645 1,269.0 265,602,250 170.7985 4.6014
040:00:00 7/18/69, 05:32:00 167.1362 77.667 1,230.5 274,391,185 171.1492 4.3906
042:00:00 7/18/69, 07:32:00 167.5194 77.677 1,194.4 282,918,557 171.4781 4.1926
044:00:00 7/18/69, 09:32:00 167.8805 77.677 1,160.4 291,199,543 171.7880 4.0058
046:00:00 7/18/69, 11:32:00 168.2218 77.666 1,128.2 299,247,650 172.0808 3.8292
048:00:00 7/18/69, 13:32:00 168.5456 77.646 1,097.7 307,074,968 172.3583 3.6616
051:00:00 7/18/69, 16:32:00 169.0018 77.600 1,054.7 318,425,471 172.7493 3.4252
054:00:00 7/18/69, 19:32:00 169.4273 77.536 1,014.7 329,335,876 173.1138 3.2046
057:00:00 7/18/69, 22:32:00 169.8260 77.456 977.1 339,834,316 173.4551 2.9978
060:00:00 7/19/69, 01:32:00 170.2012 77.360 941.8 349,945,393 173.7762 2.8031
063:00:00 7/19/69, 04:32:00 170.5556 77.249 908.4 359,690,771 174.0795 2.6191
066:00:00 7/19/69, 07:32:00 170.8917 77.124 876.8 369,089,655 174.3670 2.4445
069:00:00 7/19/69, 10:32:00 171.2114 76.984 846.8 378,159,169 174.6404 2.2784
072:00:00 7/19/69, 13:32:00 171.5164 76.830 818.1 386,914,660 174.9011 2.1199
075:00:00 7/19/69, 16:32:00 171.8081 76.662 790.7 395,369,948 175.1505 1.9683


Mapping the Trajectory

Data like that shown in the above table allows for the mapping of the trajectory. By converting the polar coordinates to rectangular (X,Y,Z) coordinates, the trajectory can be plotted graphically and shown from different perspectives. Before we do that, however, let's first take a look at what the trajectory looks like from an Earth-centered view:

This is an inside-out view showing the track of Apollo 11 across Earth's sky, indicated by the dark blue line. The light blue markers show the location of Apollo 11 in four-hour increments. The yellow disc is the position of the Sun at TLI, the gray circle is the Moon at TLI, and the gray disc is the position of the Moon at lunar-orbit insertion (075:49:50 GET). The coordinates are shown in units of right ascension and declination, which are the normal coordinates used for locating objects on the celestial sphere in the equatorial system. Declination is equivalent to latitude and right ascension is analogous to longitude with one hour equaling 15 degrees. (Note that right ascension increases right-to-left because of the inside-out perspective.) Since the viewing perceptive is geocentric, the apparent position of the spacecraft will be different for an observer on the surface of Earth due to parallax. The apparent motion of the spacecraft decreases dramatically after the first several hours due to (1) the spacecraft is slowing down and (2) it is moving mostly away from us rather than across our field of view.

Let's now examine the flight path in three dimensions. For this exercise I've defined three axes, as follows:

These axes were so defined to depict the trajectory in the correct orientation, i.e. north up and orbit counterclockwise.

The first illustration is a "Top View" looking down on Earth's north pole along the Z-axis. The path of Apollo 11 is represented by the blue line and starts at 002:50:13 GET (TLI) and ends at 075:49:50 GET (LOI). The aqua disc is Earth and the gray disc is the Moon as it was positioned at lunar orbit insertion. Both axes are drawn to the same scale with the units in kilometers.

The second illustration is a "Side View" looking along the Y-axis in the direction of 270o longitude. In this view, Earth's axis is straight up-and-down (the Z-axis) and we're viewing the equatorial plane edge-on (the X-axis). The drawing scale is the same as the Top View.

The third illustration is an "End View" looking along the X-axis in the direction of 180o longitude. Like the Side View, Earth's axis is vertical and the equator is seen edge-on. The scale here is larger than the Top and Side Views.

Together, these drawings show the translunar trajectory in all three dimensions. With some spatial visualization, it should not be too hard to see the flight path Apollo 11 followed en route to the Moon. Generally, it is an elongated ellipse inclined along the minor axis.

Something you likely noticed in these illustrations is that the trajectory misses the Moon. How is that possible? There are two reasons. First, Apollo 11 made a mid-course correction, which has been omitted from my calculations. Second, and most significantly, the gravity of the Moon has been ignored. As Apollo 11 traveled away from Earth, it started to be pulled toward the Moon, thus deflecting the trajectory. In these illustrations we see the path Apollo 11 would have followed had the Moon not been there.

To illustrate how the Moon influences the flight path, the following is a simulation of a free-return trajectory. This is just a generic illustration and does not represent any specific mission. Note how the trajectory begins to noticeably bend as the spacecraft nears the 350,000 km mark.

Free-Return Trajectory


Van Allen Radiation Belts

A common claim of the moon landing conspiracy theorists is that Apollo was impossible because the Van Allen Radiation Belts (VARB) form an impenetrable barrier to human space flight beyond low Earth orbit. Aside from the fact that the man after which the VARB are named, Dr. James Van Allen, has specifically repudiated the claim, there are several things wrong with this theory. One reason is specifically relevant to this web page - the Apollo missions didn't fly straight through the teeth of the VARB, they mostly went around them.

The Van Allen Radiation Belts are a torus of energetic charged particles circling Earth around its magnetic equator and held in place by Earth's magnetic field. The VARB are split into two distinct belts, with energetic electrons forming the outer belt and a combination of protons and electrons forming the inner belt. The energy and density of the particles varies by many orders of magnitude depending on where inside the VARB one is located.

As we have seen from the previous illustrations, Apollo 11's translunar trajectory was inclined, allowing the spacecraft to rise rapidly above the equatorial plane as it headed away from Earth. (Although we've been dealing specifically with Apollo 11, all Apollo missions flew similar trajectories.) Since we know Apollo 11's equatorial latitude and radial distance from Earth, we can easily calculate the spacecraft's horizontal and vertical distances from Earth relative to the geographic equator and axis of rotation. Apollo 11's distance from Earth in the equatorial plane is R×cos(d), and the altitude above the equator is R×sin(d). The following graph shows Apollo 11's altitude above the equatorial plane relative to its horizontal distance from Earth:

Unlike the previous illustrations, which attempted to accurately depict what the orbit looked like from various perspectives, the above does not represent any specific viewing angle. The radial movement of the spacecraft is depicted in a flattened out profile view. Motion tangential to Earth's surface is in the third dimension not visible. The red markers indicate the spacecraft position in 10-minute increments, with four hours total being shown.

The above is helpful, but it doesn't really tell us what we need to know, since the VARB circle the geomagnetic equator rather than the geographic equator. We need to transform the coordinates to yet another reference plane. To do so, we must know the inclination and ascending node of the geomagnetic plane in relation to geographic equator. In 1969, the north geomagnetic pole was located at approximately 78.6o N and 70.2o W. The inclination angle is, therefore, 90 - 78.6 = 11.4 degrees. The longitude of the ascending node is 90o ahead of the north geomagnetic pole, or -70.2o + 90 = 19.8o E.

Unlike the ascending node of the orbital plane, which is fixed in relation to the stars, the ascending node of the geomagnetic plane is fixed to Earth. As Earth rotates, the node moves through 360 degrees of celestial longitude each day, and is, therefore, time dependent. The celestial longitude of the ascending node at the time of TLI is equal to the local sidereal time: LST @ 19.8o E, 16-Jul-69, 16:22:13 GMT = 13h 18' 43" = 199.68o. The node advances at a rate of 360 / 86,164.1 = 0.00417807o/sec.

Locating Apollo 11 in relation to the geomagnetic plane is similar to the method employed earlier: (1) determine the equatorial coordinates, (2) calculate the longitude of the geomagnetic plane's ascending node, and (3) transform the coordinates using the previously given equations. After deriving the spacecraft's latitude in the geomagnetic coordinate system, we can plot Apollo 11's horizontal and vertical distances from Earth relative to the geomagnetic equator and axis, as follows:

Apollo 11's TLI launch point was very close to the descending node of the geomagnetic plane, which is a very advantageous place to start. As the spacecraft swings around Earth and heads out toward the Moon, it travels in the direction where the geomagnetic plane slopes away from it. In fact, by the time Apollo 11 reaches a distance of about three Earth radii, the geomagnetic axis is tilted almost exactly in the direction of the spacecraft, resulting in maximum separation between Apollo 11 and the geomagnetic plane. This optimal alignment is maintained until the spacecraft is well beyond the limits of the VARB.

Let's now overlay the trajectory on flux maps of the VARB (resizing to match scale). The first map shows the electron flux and the next two maps show the proton flux at two energy levels. As you can see, Apollo 11 easily avoided the areas of highest flux, thereby minimizing the exposure. As before, the red markers indicate the time in 10-minute increments. The far edge of the electron belt was reached in about 90 minutes, the inner zone was traversed in about 30 minutes, and the region of the most energetic particles was skirted in just about 10 minutes.


From AP-8 Trapped Proton Environment for Solar Maximum and Solar Minimum, National Space Science Data Center, December 1976.


From AP-8 Trapped Proton Environment for Solar Maximum and Solar Minimum, National Space Science Data Center, December 1976.

Below is another VARB map showing the radiation exposure rate in rads per second. We see clearly that Apollo 11 bypassed the most intense zones, instead skirting along the edges. Furthermore, in agreement with the electron belt overlay, we see Apollo 11 leaving the vicinity of the VARB after about 90 minutes.

Of course, the outbound leg was only half the journey; Apollo 11 had to also fly past the VARB on the return trip. Applying to the transearth trajectory the same computational methods used to analyze the translunar trajectory, we can determine Apollo 11's position relative to the VARB (see below). Like before, the trajectory was optimized to bypass the VARB to the greatest extent possible. This time, however, Apollo 11 approached from the south and was traveling at an even greater inclination and velocity than it had on its way to the Moon. The spacecraft contacted the fringes of the VARB for a period of about 60 minutes.

From the above we can estimate the theoretical radiation dose received; however, we must do some extrapolation because, for a time, Apollo 11's flight path strayed outside the mapped area. I've done my best to estimate how long Apollo 11 was in each of the colored zones, thereby allowing a calculation of the radiation dose. Be advised, however, that the above map is just an average representation of the VARB. In fact, the VARB are not uniformly distributed around Earth, being compressed on the sunward side and elongated on the opposite side. Furthermore, the radiation intensity varies depending on solar activity. The actual geometry and intensity of the VARB at the time and direction of Apollo 11's flight may be different than that shown.

Estimated Radiation Dose
Zone Duration (min) Rate
(Rad/s)
Dose
(Rads)
Outbound Inbound
Blue 53 33 0.0001 0.52
Green 17 15 0.001 1.92
Yellow 10 10 0.005 6.00
Orange 5 0 0.01 3.00
TOTAL 11.44

Rad is a unit of absorbed dose, while rem is a unit of dose equivalent and its biological effects in man. The dose equivalent in rems is equal to the absorbed dose in rads multiplied by a quality factor, Q, for the specific type of radiation. For beta radiation (electrons) Q = 1, and for high-energy protons Q = 10. The VARB consists of both electrons and protons, with electrons forming the outer belt and a combination of protons and electrons forming the inner belt. As can be seen from the above, most of the dose is received from the outer belt, i.e. from beta radiation**. It is impossible to accurately determine the dose equivalent from the limited data provided here, but a reasonable guess might place the value at somewhere around 20-25 rem (Q ≈ 2). A person will experience radiation sickness with a dose of 100-200 rem, and death with a dose of 300+ rem.

This is, of course, the radiation dose that an unprotected astronaut, i.e. one outside the spacecraft, would receive. The Apollo 11 astronauts where safely inside a spacecraft having a multi-layer hull with a total thickness of several centimeters (see below) and a shielding rating of 7 to 8 g/cm2. This hull provided excellent shielding from the particulate radiation of the VARB, as it was constructed from materials ideal at stopping this type of radiation. The actual radiation dose received by the astronauts throughout the entirety of the Apollo 11 mission was less than one rem, as measured by dosimeters.

(** Please note that this is true only for the case of the unprotected astronaut, as beta radiation lacked the energy necessary to penetrate the spacecraft shielding. It was the highest energy protons of the inner radiation belt that posed the only real concern in trajectory planning and, as we have seen, Apollo was beyond this region in as little as ten minutes.)


From Virtual Apollo, Apogee Books, Scott P. Sullivan.