Buidling a Miniature Solar System Part 2

H is for Hyperspace

 

ohmannExpress:
When it Absolutely, Positively Has To Be There Overnight

The Hohmann Transfer Orbit is one of the most delta-V efficient transfer orbits
there is but the efficiency comes at the price of having to use very precise
departure and arrival times as well as very long travel times. (There are more
efficient orbits but they have even longer travel times and often require
gravitational slingshot maneuvers.)

Efficiency is important because even in the 21st century, we are stuck using
wimpy chemical rockets that make you watch every milligram of payload. Even
atomic rockets try to use Hohmann transfers orbits or suffer large payload mass
penalties.

Calculating Hohmann orbits is also useful as a measuring rod to illustrate the tradeoffs
you have to make between different types of rockets, mass ratios and flight
times.

It’s not like the good old days where Flash, Buck, Tom or Dan just jumped into their
rockets, lit the candle and a short while later crashed into Venus, Mars or
Mongo. The average Hohmann transfer time going one way between Earth and Mars
is about 9 months. Being packed like spam in a can for months on end is, let’s
face it, boring. Unless your crewmate is a homicidal maniac, flying between the
planets isn’t as exciting as most eager young space cadets think.

That’s the reality.

Basically, reality stinks

Quick philosophical question. What is reality?

Answer. Reality is what you can get away with.

If you can’t do something one way, try another. If you have to, change the
conditions so you can do what you want. (I learned that in a movie) In other
words, cheat.

Most science fiction stories cheat by changed the way gravity works or just by
ignoring the huge amount of energy required to move stuff around quickly in
space. It’s an understandable cheat, but everyone does that.

I want to do something different.

I want to make a really, really small solar system that is easier to get around in.
Something that doesn’t take month or years to move around in. I’m calling it the Ognom System.

Ognom is a large Gas Giant in the habitable zone of a star. It has several very large
planet sized moons orbiting it. Some of which are life bearing.  The basic data about the Ognom System is in the
following tables.

test22

Ognom Mass KG

2.079E+28

Ognom Radius KM

70000

Ognom Volume KM^3

1.436755E+15

Ognom Density KG/M

1.447E+05

Here is the basic data for five of the major moons in orbit around Ognom.

test11A

test6A

test8A

TEST1A

test15A

Name

Kalmor

Palenmor

Malmor

Balanmor

Xalamor

SemiMajor Axis
KM

450257

658709

1045635

1370169

2175008

SemiMajor Axis
AU

3.01E-03

4.40E-03

6.99E-03

9.16E-03

1.45E-02

Eccentricity

0.00200

0.00100

0.00150

0.00200

0.03000

Inclination

0.03

0.2

2

3

5

Orbital Period
Days

0.5898

1.0437

2.0874

3.1311

6.2622

Orbital Velocity
(KM/S)

55.51349

45.896772

36.428292

31.82304

25.25797

Radius KM

1821.6

5285

3430

4025

3750

Radius
(Earth = 1)

0.285607

0.8286297

0.5377861

0.631076

0.587959

Diameter KM

3643.2

10570

6860

8050

7500

Circumference
KM

11445.45

33206.634

21551.326

25289.82

23561.94

Surface Area
KM^2

4.17E+07

3.51E+08

1.48E+08

2.04E+08

1.77E+08

Surface Area
(Earth = 1)

0.08

0.69

0.29

0.40

0.35

Mass KG

1.32E+23

2.50E+24

6.57E+23

1.17E+24

1.10E+24

Mass (Earth=1)

0.02

0.51

0.11

0.20

0.18

Volume

2.53E+10

6.18E+11

1.69E+11

2.73E+11

2.21E+11

Density Kg/M^3

5202

4050

3887

4273

4971

Surface Gravity
Earth 1G

0.27

0.61

0.38

0.49

0.53

Surface Gravity
M/S

2.649

5.983

3.727

4.808

5.211

Escape Velocity
KM/S

3.11

7.95

5.06

6.22

6.25

DeltaV Liftoff
to Orbit M/S

2196.48

5622.83

3575.17

4398.59

4420.24

DeltaV Liftoff
to Orbit KM/S

2.20

5.62

3.58

4.40

4.42

Since I am interested in quickly traveling between worlds, the first question I want to
ask is how long does it take to get from, lets say Malmor to Palenmor?

The equation for the transfer times for Hohmann transfer orbits is:

Hohmanntransfer

Where:

tH= Hohmann transfer time
π = pi = 3.14159265359
a = Semi-Major Axis of the Hohman Orbit (Has to be calculated. We don’t have
this yet)
r1 = Radius of the Interior (Or Inferior) Planet’s Orbit
r2 = Radius of the Exterior (Or Superior) Planet’s Orbit
µ = Standard Gravitation Parameter which is found by multiplying the universal gravitation constant times the mass of the primary body. In this case Ognom is the primary body and its mass is 2.079E+28 kg. So the standard gravitational parameter is: 6.67348E-11 m3 kg-1 s-2     *  2.079E+28 kg  =  1.388E+18  m3 s-2

We’re going to use the second equation that just uses the orbital radius of the two planets. To get the transfer time, insert the orbital radius (or semi-major axis) of the planet you are departing from and the one you are arriving at to find the length of time it takes to travel between the two worlds.

Worked example: Traveling from Malmor to Palenmor

π * sqrt(((658709KM + 1045635KM)3)/(8*1.388E+18 m3 s-2)
= 66344.832 seconds
= 1105.7472 minutes
= 18.42912 hours
= .76788 days

HohmannOrbit1

18 and a half hours travel time from Malmor to Palenmor. Not bad. It’s a heck of a lot better than the 9 month average to get from Earth to Mars.

Finding the one way travel times between the other worlds is left as an exercise for the reader.

Traveling using a Hohmann transfer orbit requires that some specific conditions be met before you can start. You can only do this when the two planets are aligned in the correct position. If they are out of alignment you can go ahead and rocket off but the planet you are wanting to visit won’t be there to meet you.

Since we now know how long it takes to get from our departure and destination planet we now need to find what position your destination planet has to be at when you depart..

To do this we try and find the phase angle of departure between the two planets. The phase angle is made by forming a triangle between the current position of planet A, the central mass, or orbital focus, and the current position of planet B.

The easy way to calculate this is to use the following equation.

phaseangle

r1 = Radius of the departure Planet’s Orbit
r2 = Radius of the destination Planet Orbit

So for our worked example we can plug in the numbers to find that the phase angle needs to be

180 * ((1- (658709KM + 1045635KM)/(2*658709KM))1.5) = -84.864 degrees

What this means is that the destination planet needs to be 84.864 degrees behind the departure planet when viewing their orbits from above. When the calculation gives you a negative angle then the destination planet needs to be trailing behind
the departure planet as they travel counter clockwise. If the calculation gives you a positive angle then the destination planet needs to be ahead of the departure planet.

Basically finding the phase angle is just a fancy, but exact way to try  to hit a moving target. Basically you are just leading your target. When shooting at a moving target, you don’t aim at where the target currently is, you aim at where the target will be when your shot arrives. The phase angle gives you the proper amount of lead for the shot so you can hit the target.

Take a look at the diagram below.

HohmannOrbit2

At a start time, lets call it t1, Malmor and Palenmor are in the correct position so you can use a Hohmann transfer orbit to travel from Malmor to Palenmor. At time t1, Palenmor is on the left side of the diagram and Malmor is at the bottom. The phase angle
between Palenmor and Mallmor is -84.864 degrees. Since Palenmor is in the interior orbit, it’s orbital velocity is faster then Malmor’s. As times passes, Palenmor will move ahead of Malmor until it reaches it’s new position toward the top of the diagram at our arrive time t2. The difference between times t1 and t2 is the same as the Hohmann transfer time of 18.42912 hours. In 18.5 hours Palenmor moves around almost ¾’s of it’s orbit. In the same amount of time Malmor moves through almost 1/3rd of it’s orbit.  You, traveling in your rocketship, move half way through your Hohmann orbit until you arrive at your destination.

The next thing we need to know is how often do the planets to line up in this position.  The amount of time this takes is called the Synodic Period. To find this we use this equation.

synodic

p1 = Period of the Departure Planet’s Orbit
p2 = Period of the Destination Planet’s Orbit

For our worked example the synodic period is

1/((1/1.0437) – (1/2.0874)) = 2.0874 days

That means we can use the Hohmann orbit in a little over  every two days to get from Malmor to Palenmor.

To summarize, if you miss your first change to go from Palenmor to Malmor you have
to wait a little over 2 days before the planets to align again so you can start
on you flight to Palenmor.

So compared to getting from earth to mars, arriving in less then 3 days ain’t bad.

Below is a table of Hohmann transfer information for moons of Ognom.

To Kalmor
From Travel Time Days Phase Angle Synodic Days
Palenmor 0.40 -65.99 1.36
Malmor 0.63 -205.38 0.82
Balanmor 0.85 -337.36 0.73
Xalamor 1.47 -715.98 0.65
To Palenmor
From Travel Time Days Phase Angle Synodic Days
Kalmor 0.40 40.98 1.36
Malmor 0.77 -84.86 2.09
Balanmor 1.00 -164.01 1.57
Xalamor 1.65 -387.84 1.25
To Malmor
From Travel Time Days Phase Angle Synodic Days
Kalmor 0.63 71.11 0.82
Palenmor 0.77 47.57 2.09
Balanmor 1.30 -43.49 6.26
Xalamor 1.99 -164.01 3.13
To Balanmor
From Travel Time Days Phase Angle Synodic Days
Kalmor 0.85 82.54 0.73
Palenmor 1.00 65.33 1.57
Malmor 1.30 31.01 6.26
Xalamor 2.30 -84.86 6.26
To Xalamor
From Travel Time Days Phase Angle Synodic Days
Kalmor 1.47 95.61 0.65
Palenmor 1.65 85.36 1.25
Malmor 1.99 65.33 3.13
Balanmor 2.30 47.57 6.26

By looking at the table, the shortest travel time is less than half a day, the longest is less than two and a half days. Synodic Period range from .65 days to a little over six and a quarter days.

Finally we need to know that amount of delta v, or change in velocity, we need to make to move from one orbit to another. For this I am assuming we need to make two rocket burns break orbit on departure and then brake at our destination. The equations for the total delta-v using two instantaneous rocket impulses for departure and arrival is:

deltav1

deltav2

deltavtotal

Δv1 = Change in velocity when we depart.
Δv2 = Change in velocity when we arrive.
Δvtotal = Total change in velocity

r1 = Radius of the Interior (Or Inferior) Planet’s Orbit
r2 = Radius of the Exterior (Or Superior) Planet’s Orbit
µ = Standard Gravitation Parameter = 1.388E+18 m3 s-2

HohmannOrbit1

Δv1 for our example is found by

sqrt(1.388E+18 m3 s-2 / 658709KM) * (sqrt((2*1045635KM)/( 658709KM + 1045635KM)) – 1)
Δv1 = 4.40087 KM/S

Δv2 for our example is found by

sqrt(1.388E+18 m3 s-2 / 1045635KM) * (1 – sqrt((2*658709KM)/( 658709KM + 1045635KM)))
Δv2 = 4.94358556 KM/S

Δvtotal is therefore:
Δvtotal = 4.40087 + 4.94358556 = 9.344459 KM/S

So any spaceship we use has to be capable of changing it velocity by a total of 9.34 KM/S. This is actually more then the amount of Δv you need to get from Earth to Mars. So that is another trade off we have to deal with by making this miniature planetary system.

For more details on orbital mechanics and rocketing around space I highly recommend the Atomic Rockets and the Basics of Space Flight websites.

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