The First Orbit

Semi-major axis:

a = R_earth + (perigee_altitude + apogee_altitude) / 2

Kepler’s third law, in a form that gives you the period directly:

T = 2π · √(a³ / μ)

That’s in seconds. There are 60 of them in a minute.

Semi-major axis, from the center of the Earth:

a = R_earth + (perigee_altitude + apogee_altitude) / 2

Kepler’s third law:

T = 2π · √(a³ / μ)

Answer: ≈ 96.21 minutes — matches the historically reported period almost exactly.

215    ENTER
939    +
2      /
6378.137  +
3      Y^X
398600.4418  /
SQRT
2      *
PI     *
60     /

Display: 96.209568

One habit worth having on this machine: don’t use x^y (F xy) to cube a number this large. The real hardware computes x^y via an internal log/exponential routine with a limited range, and it genuinely errors out on a base in the thousands — this isn’t a simulator quirk, it’s the real 1970s chip. Cube by plain multiplication through a memory register instead.

The store/recall keys are two-key sequences, and the labels don’t say “STO”/“RCL” the way HP keyboards do: the store key is marked x→П (x into memory), the recall key П→x (memory into x). They’re easy to mix up since they use the same two glyphs in opposite order — pressing П→x when you mean x→П recalls from an empty register instead of writing to it.

√ and π below are both F-shifted keys: F - for √x, F + for π.

215    ENTER
939    +
2      /
6378.137  +
x→П 0          store (x→П, then 0)
ENTER
×              a²
П→x 0          recall (П→x, then 0)
×              a³
398600.4418  /
√x
2      ×
π      ×
60     /

Display: 96.209565

Same structure as the 42S; the 15C has y^x on its own key.

215    ENTER
939    +
2      ÷
6378.137  +
3      y^x
398600.4418  ÷
√x
2      ×
π      ×
60     ÷

Result: ≈ 96.2096 minutes.

215    ENTER
939    +
2      /
6378.137  +
3      Y^X
398600.4418  /
SQRT
2      *
PI     *
60     /

Display: ≈ 96.2096 minutes.

Plain stack entry, the same shape as the 42S — with one RPL-specific sting in the tail. √x and y^x are unshifted keys here, and π is ← SPC.

The catch is that π doesn’t push a number. It pushes the symbolic constant 'π', and with system flags −2 and −3 in their default cleared state a symbolic constant stays symbolic all the way through the arithmetic. Run the sequence below and stop at the last ÷ and level 1 holds '1837.46739429*π/60' — an exact answer, algebraically, and useless if what you wanted was a number of minutes. →NUM (← EVAL) is what collapses it:

215    ENTER
939    +
2      ÷
6378.137  +
3      y^x
398600.4418  ÷
√x
2      ×
π      ×
60     ÷
→NUM

Result: 96.2095677852 minutes, matching the HP-42S value to eleven significant digits — the last digit or two is the 48’s real twelve-digit BCD arithmetic, not a different answer.

This is the one place on the site where RPL’s habit of staying symbolic changes what you actually have to press, which is a fair answer to the question of what an RPL machine does differently.

A much sparser keyboard than the later machines: there are no dedicated y^x, √x or π keys. y^x and √x are gold f functions on the 3 and 2 keys, and π is a blue g function on the • key. Engineering mode (f ENG 1) keeps the exponent in view at every step:

f ENG 1                  0.000   00
939      ENTER          939.0   00
215        +            1.154   03
2          ÷            577.0   00
6378       +            6.955   03
3      f  y^x           336.4   09
398600     ÷            844.0   03
       f  √x            918.7   00
g π        ×            2.886   03
30         ÷            96.21   00

Two things worth stealing from this one. The last step divides by 30 rather than 60, with no separate × 2, because 2π ÷ 60 is the same number as π ÷ 30. And the radius used is a flat 6378 rather than 6378.137, which is the sort of rounding a four-digit display invites: it moves the answer to 96.2068 against the 42S’s 96.2095677850, and at the precision shown both read the same.

Result: 96.21 minutes. The four significant digits are the ENG 1 display setting, not the limit of the machine — an HP-25 carries ten digits internally, and f SCI 9 or a FIX setting will show you more of them.

2π√(((215+939)/2+6378.137)^3/398600.4418)/60

Result: ≈ 96.2096 minutes.

Same single expression, natural display mode:

2π√(((215+939)÷2+6378.137)³÷398600.4418)÷60

Result: ≈ 96.2096 minutes.

The only addition in the whole calculation is finding the semi-major axis — averaging perigee and apogee, then adding Earth’s radius. That’s pencil work, exactly like the unit conversions in “The Manual Burn”: do it on paper before you touch the rule, then everything left is multiplication, division, a cube, and a square root, all of which the C/D/K/A scales handle directly.

1. Pencil step: a = (215 + 939)/2 + 6378.137 ≈ 6955.14 km.
2. Cube a using the K scale against D: set the cursor on 6.955 on the D scale (tracking the exponent separately, 6955 = 6.955×10³), and read the cube directly off K: ≈ 336, giving a³ ≈ 336×10⁹.
3. Divide by μ (398,600 ≈ 3.986×10⁵) on the C/D scales: set the cursor on 3.36 (D), bring 3.986 on C under the cursor, and read the result at C’s left index: ≈ 0.844×10⁶ = 8.44×10⁵.
4. Take the square root on the A/B (or D/A) scale — an even power of ten, so it stays clean: √(8.44×10⁵) ≈ 9.19×10² = 919.
5. Multiply by 2π (use the CI scale if your rule has one, to fold the ×2 and ×π into one setting): 919 × 2π ≈ 5773.
6. Divide by 60 to get minutes: 5773 / 60 ≈ 96.2.

Answer: ≈ 96.2 minutes — three significant figures is a slide rule’s honest precision here; hand-checked against the verified 96.2096 result above.