16.50 Propulsion Systems
Spring 2012
Homework 2: Chemical vs. Electrical Thrusters
a) Chemical:
We start outside the sphere of influence (SOI) of Earth, and end outside the SOI of Mars, so no “escape”
or “capture” οܸ‫ ݏ‬are involved. The whole motion is under the Sun’s influence alone.
To enter the transfer orbit:
ఓೄ
οܸଵ ൌ ‫ݒ‬௣௘௥௜௛௘௟௜௢௡ െ ‫ݒ‬௖ǡா௔௥௧௛ ൌ ට
௥ಶ
൬ට
௥
ଶ௥ಾ
ಶ ା௥ಾ
െ ͳ൰
(1)
Known and calculated values:
ߤௌ ൌ ͳǤ͵ʹ͹‫Ͳʹܧ‬
௠య
௦మ
‫ݎ‬ா ൌ ͳǤͶͻ͸‫݉ͳͳܧ‬
‫ݎ‬ெ ൌ ͳǤͷʹ͵͹ ‫ݎ כ‬ா ൌ ʹǤʹ͹ͻ‫݉ͳͳܧ‬
௠
‫ݒ‬௖ǡா௔௥௧௛ ൌ ʹͻǡ͹ͺͲ
‫ݒ‬௖ǡெ௔௥௦ ൌ
௦
௠
ʹͶǡͳ͵Ͳ
௦
Substituting values:
ଶ‫כ‬ଵǤହଶଷ଻
οܸଵ ൌ ʹͻǡ͹ͺͲ ቆට
ଶǤହଶଷ଻
െ ͳቇ ൌ ʹǡͻͶͷ
௠
௦
To enter circular orbit near Mars:
ఓ
οܸଶ ൌ ‫ݒ‬௖ǡெ௔௥௦ െ ‫ݒ‬௔௣௢௛௘௟௜௢௡ ൌ ට ೄ ൬ͳ െ ට
௥
௥
ଶ
ቇ
ଶǤହଶଷ଻
οܸଶ ൌ ʹͶǡͳ͵Ͳ ቆͳ െ ට
ൌ ʹǡ͸Ͷͻ
ଶ௥ಶ
ಶ ା௥ಾ
ಾ
൰
(2)
௠
௦
Total οࢂ:
1
οܸ ൌ οܸଵ ൅ οܸଶ ൌ ͷǤͷͶͻ
௠
(Chemical)
௦
Transfer duration:
The transfer time is ½ the orbital time in the transfer ellipse.
௥ಾ ା௥ಶ
ൌ ͳǤͺͺ͹͹‫݉ͳͳܧ‬
ଶ
ଵ
௔యȀమ
‫כ ߨʹ כ‬
(3)
ଶ
ඥ ఓೄ
ଶǤଶଷ଻ா଻
ʹǤʹ͵͹‫ܧ‬͹‫ ݏ‬ൌ
ൌ ʹͷͻ݀ܽ‫ݏݕ‬
଼଺ସ଴଴
Semiaxis: ܽ ൌ
ο‫ ݐ‬ൌ
ο‫ ݐ‬ൌ
ࡹ࢖ࢇ࢟
ࡹ૙
οೇ
ൌ ࢋି ೎ െ ߝ
(4)
ఱǡఱరళ
‫ܯ‬௣௔௬ ൌ ʹͲǡͲͲͲ ൬݁
ିరǡఱబబ
൰ െ ͲǤͲͷ ൌ Ͷǡ͹͹Ͳ݇݃
b) Electrical:
The propulsive οࢂ is now:
οܸ ൌ ‫ݒ‬௖ǡா െ ‫ݒ‬௖ǡெ ൌ ʹͻǡ͹ͺͲ െ ʹͶǡͳ͵Ͳ ൌ ͷǡ͸ͷͲ
௠
௦
(Electric Propulsion)
This is only slightly more than the chemical οܸ; for transfers to larger radii, the difference is more
noticeable.
For optimization of the low-thrust mission, define non-dimensional variables:
ߤൌ
ெ೛ೌ೤
(5)
ெబ
‫ݒ‬ൌ
ο௏
௖
(6)
ߣൌ
ఈ௔బ ο௏
ଶఎ
(7)
ߝൌ
ெೞ೟ೝ
ெబ
(8)
ߙ ൌ ͳͲ
௞௚
௞ௐ
ൌ ͲǤͲͳ
௞௚
ௐ
is the specific mass (per unit power) of the power and propulsion equipment.
ܽ଴ is the initial acceleration.
Combining expressions:
ఒ
௩
ߤ ൌ ݁ ି௩ െ െ ߝ
(9)
To find the best specific impulse ܿ, we have differentiate with respect to ‫ݒ‬:
െ݁ ି௩ ൅
ఒ
௩మ
ൌͲ
2
ߣ ൌ ‫ ݒ‬ଶ ݁ ି௩
(10)
Substituting values:
ߣൌ
଴Ǥ଴ଵ‫כ‬ହ଺ହ଴
ଶ‫כ‬଴Ǥ଻
ൌ ͶͲǤ͵ͻܽ଴
For each value of ܽ଴ we then need to solve (by trial and error) the equation:
ଶ
ͶͲǤ͵ͻܽ଴ ൌ ‫ݒ‬௢௣௧
݁ ି௩೚೛೟
(11)
Once ‫ݒ‬௢௣௧ is known, we calculate:
ܿ௢௣௧ ൌ
୼୚
௩೚೛೟
ൌ
ହ଺ହ଴
௩೚೛೟
The implied transfer time follows from:
ష౴ೇ
ȟ‫ ݐ‬ൌ
ఓ೛ೝ೚೛
௠ሶ
ெబ ቆଵି௘ ೎ ቇ
ൌ
ிൗ
஼
ൌ
௖
൬ͳ
௔బ
౴ೇ
െ ݁ି ೎ ൰ ؆
ο௏
ο௏
݂݅
௔బ
௖
‫ܿا‬
(12)
The power per unit initial mass:
௉
ெబ
ൌ
ி௖ൗ
ଶఎ
ெబ
ൌ
௔బ ௖
ଶఎ
(13)
Finally, the payload mass is:
ߤ௣௔௬ ൌ ߤ௢௣௧ ‫ܯ‬଴ ൌ ‫ܯ‬଴ ൬݁ ି௩೚೛೟ െ
ఒ
௩೚೛೟
െ ߝ൰
(14)
The results are tabulated below for a range of initial accelerations:
ܽ଴ ቂ݉ൗ ଶ ቃ
‫ݏ‬
ߣ
‫ݒ‬௢௣௧
ܿ௢௣௧ ሾ݉Τ‫ݏ‬ሿ
ȟ‫ݐ‬ሾ݀ܽ‫ݏݕ‬ሿ
633.3
322.2
(chem)
152.9
100.3
1E-4
2E-4
4.039E-3
8.078E-3
0.06567
0.09421
86052
60004
4E-4
6E-4
0.01616
0.02423
0.1361
0.1694
41536
33371
ܲ ܹ
‫ܯ‬௣௔௬ ሾ݇݃ሿ
ቂ ൗ݇݃ቃ
‫ܯ‬଴
6.149
16498
8.572
15486
0.123
0.1714
11.867
14.302
0.2373
0286
14082
13024
We see several important things here:
a) For any specific power ܲൗ‫ ܯ‬൒ ͳͲ ܹൗ݇݃ , the transfer is faster than chemical.
଴
b) The payload delivered is 3-4 times greater than with chemical.
c) The specific impulse is in the range from 3,400s to 8,700s.
3
ܲሾ‫ܹܯ‬ሿ
d) The required power is from 120-290 KW, possible with large solar arrays.
4
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