Newton solution of Keplerian equation
(1) M=MA*2*3.14/256 : Radian from MA
(2) to=M＋e*sin(M)＋0.5*e^2*sin(2*M) : Initial value
(3) mo=to-e*sin(to) : Keplerian equation
(4) dto=(M-mo)/(1-e*cos(to)) : e=Eccentricity
(5) t1=to+dto : 1st approximate value
(6) m1=t1-e*sin(t1) : calculation
(7) dt1=(M-m1)/(1-e*cos(t1)) : calculation
(8) t2=t1+dt1 : 2nd approximate value
(9) m2=t2-e*sin(t2) : calculation
(10) dt2=(M-m2)/(1-e*cos(t2)) : calculation
(11) t3=t2+dt2 : 3rd approximate value
(12) t4=tan(t3/2) : calculation
(13) u=root((1+e)/(1-e))*t4 : calculation
(14) s=2*arctan(u) : Real Anomaly
(15) p=24*3600/n : Period (total seconds)
(16) 4*3.14^2*(a^3/p^2)=G*Q : A law of the universal graviation
(17) a=(((G*Q*p^2)/(4*3.14^2))^(1/3))*10^9 : Semi Major Axis
(18) r0=a(1-e^2)/(1+e*cos(s)) : Altitude from the center of the earth
(19) r=r0-637814000 : Altitude from the face of the earth
Supplement of (16),(17):
a = (((G*Q*p^2)/(4*3.14^2))^(1/3))*10^9
a = (6.6732*10^(-8)*5.9732*10^27*(p*p)/(4*3.14159*3.14159))^(1/3)
Then, G is the constant of the universal gravitation,
Q is the mass of the earth.
[N] = [(kg*m)/s^2] = 10^3 * 10^2 [(g*cm)/s^2]
= 10^5 [(g*cm)/s^2]
[m^2] = 10^4 [cm^2], [kg^2] = 10^6 [g^2]
G = 6.6732 * 10^(-11) [N*m^2/kg^2]
= 6.6732 * 10^(-11) * 10^5 * 10^4 / 10^6 [cm^3/(g*(s^2))]
= 6.6732 * 10^(-8) [cm^3/(g*(s^2))]
Q = 5.9732 * 10^24 [kg]
= 5.9732 * 10^27 [g]
R : Mass of the satellite
a : Semi Major Axis
p : Period
Then, R * a * (2π/p)^2 = (G * Q * R) / a^2
i.e. p^2 = (4π^2 / (G * Q)) * a^3
so, a = ((G * Q) * p^2 / 4π^2 )^(1/3)
The unit is [(cm^3/(g*(s^2)) * g * s^2)^(1/3)] = [cm].
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Mean Anomaly Eccentri. Ini.val. Kep.equation 1st appro.value 2nd appro.value 3rd appro.value Real anomaly Mean motion Period (h) (m) SMA Dist.C Dist.S
MA MA' M e to Mo dto t1 m1 dt1 t2 m2 dt2 t3 t4 u s n p p1 p2 a r rh
(sample1 : 16 Jan 1994, AO-13)
22 30.9 0.53969 0.72099 1.139 0.484 0.079 1.218 0.541 -0.002 1.216 0.540 0.000 1.216 0.696 1.729 2.093 2.09721276 41198 11 27 2578169623 1933173674 12953.6
(sample2 : 28 May 2001, AO-40)
0.01 0.0 0.00025 0.81492 0.001 0.000 0.000 0.001 0.000 0.001 0.002 0.000 0.001 0.003 0.002 0.006 0.012 1.27026844 68017 18 54 3601408456 666581751 287.7
128 180.0 3.14000 0.81492 3.140 3.139 0.001 3.141 3.141 -0.001 3.140 3.139 0.001 3.141 3374.653 10567.542 3.141 1.27026844 68017 18 54 3601408456 6536251656 58984.4
255.99 360.0 6.27975 0.81492 6.275 6.282 -0.010 6.265 6.280 -0.001 6.264 6.280 -0.001 6.263 -0.010 -0.031 -0.062 1.27026844 68017 18 54 3601408456 667135754 293.2
(sample3 : 23 Jun 2001, AO-40 before arcjet)
0.01 0.0 0.00025 0.81508 0.001 0.000 0.000 0.001 0.000 0.001 0.002 0.000 0.001 0.003 0.002 0.006 0.012 1.2711484 67970 18 53 3599749208 665697946 278.8
86 121.4 2.10969 0.81508 2.517 2.040 0.042 2.559 2.111 -0.001 2.558 2.109 0.000 2.558 3.329 10.430 2.950 1.2711484 67970 18 53 3599749208 6046194040 54083.8
128 180.0 3.14000 0.81508 3.140 3.139 0.001 3.141 3.141 -0.001 3.140 3.139 0.001 3.141 3374.653 10572.585 3.141 1.2711484 67970 18 53 3599749208 6533816936 58960.0
(sample4 : 23 Jun 2001, AO-40 after arcjet)
0.01 0.0 0.00025 0.81501 0.001 0.000 0.000 0.001 0.000 0.001 0.002 0.000 0.001 0.003 0.002 0.006 0.012 1.2710861 67973 18 53 3599855129 665944691 281.3
100 140.5 2.45313 0.81501 2.645 2.257 0.114 2.759 2.455 -0.001 2.758 2.453 0.000 2.758 5.150 16.132 3.018 1.2710861 67973 18 53 3599855129 6321346299 56835.3
128 180.0 3.14000 0.81501 3.140 3.139 0.001 3.141 3.141 -0.001 3.140 3.139 0.001 3.141 3374.653 10570.598 3.141 1.2710861 67973 18 53 3599855129 6533782042 58959.7
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The solution of this algorithm corrected by W3IWI and JE9PEL :
Before After
Perigee height : 278.8 281.3 ....... +2.5 km
Apogee height : 58960.0 58959.7 ....... -0.3 km
The report of Peter Guelzow, DB2OS :
Before After
Perigee height : 279.754 282.2427 .... +2.5 km
Apogee height : 58971.166 58971.024 .... -0.1 km
Excel sheet of the Satellite Altitude analysis,
http://www.ne.jp/asahi/hamradio/je9pel/satrange.zip
> Date: Sat, 23 Jun 2001 10:08:15 +0000
> From: Peter Guelzow
> To: amsat-bb, amsat-dl
> Subject: AO-40: more successful ARCJET operations
>
> Dear All,
>
> The arc-jet thruster has been invoked on Orbit 296, MA 118-135,
> which gave about 1 hour of thrusting gas only.
>
> The S2 TX is OFF from MA 100-180 to spare power for the ATOS.
> The gas generator for the ammonia draws about 120 - 130 W of power
> when cycled on by the thermostat.
>
> The IHU-2 is running and logging telemetry into a circular buffer
> and hold about 2.5 days worth of data. Downloaded telemetry from
> this 1h burn indicated positive power budget and everything looks
> nominally.
>
> The thrust on orbit 296 started at MA 121.4 and lasted for 3618s.
> The acceleration is guestimated to be 54E-6 m/s^2 ,
> and the direction of acceleration is towards alon 274, alat -2
> (the current attitude).
>
> Give or take the unknowns, the expected outcome of this was:
>
> Before After
> --------------------------------------------
> Epoch year 2001 2001
> Epoch time 173.12145 173.16312
> Inclination 5.2833 5.2833592
> R.A.A.N 180.71591 180.70361
> Eccentricity 0.815077 0.8150139
> Arg perigee 288.69088 288.71333
> Mean Anomaly 121.4 140.46836
> Mean motion MM 1.2711484 1.2710861
> Revolution 296 296
> SMA 36003.6 36004.773
>
> Perigee height 279.754 282.2427 ... +2.5 km
> Apogee height 58971.166 58971.024
> --------------------------------------------
>
> So the perigee was raised by this 1h burn by about +2.5 km, but only
> a thrust rate of 50% was used.
>
> Since everything went so well, the computer onboard AO-40 was
> commanded to initiate 2h "burns" starting around apogee on orbit 297
> for the next three orbits.
>
> The first 2-hour burn stopped at Orbit 297 MA 142, which is 2001
> Jun 23 0154 utc. Again, all telemetry is looking good and we may soon
> expand to 4h "burns" and possibly increase the thrust level as well.
>
> While we haven't yet seen the latest NORAD data, some effects of the
> changing orbit should be soon noticeable.
>
> Indeed, we all are very happy with the successful results of the ATOS
> (Arcjet Thruster on OSCAR Satellite) so far.
>
> Some more information about the ATOS system is available here:
>
> http://www.irs.uni-stuttgart.de/RESEARCH/EL_PROP/PROJ/e_atos.html
>
> best 73s
> Peter DB2OS for the AO-40 team
> Date: Sun, 24 Jun 2001 18:08:22 -0400
> From: Tom Clark
> To: Mineo Wakita, amsat-bb
> Subject: Re: AO-40: more successful ARCJET operations
>
> Mineo -- Nice work. The perigee difference is probably due to your
> definition of altitude, i.e. distance from the surface of the earth.
> You said:
>
> > (18) r0=a(1-e^2)/(1+e*cos(s)) : Height from the center of the earth
> > (19) r=r0-6371 : Height from the face of the earth
>
> The equatorial radius of the oblate earth is 6378.160 km, larger than
> the 6371 km you use. Since the inclination is only 5.3 deg, it is
> appropriate to use the equatorial value.
>
> Therefore, using your formulation, I come up with:
>
> > The solution of this algorithm [posted by JE9PEL]
> > Before After
> > Perigee height : 286.0 288.4 ....... +2.4 km
> >
> > The report of Peter Guelzow, DB2OS
> > Before After
> > Perigee height : 279.754 282.2427 .... +2.5 km
>
> The JE9PEL perigee corrected by W3IWI:
> Before After
> Perigee height : 278.8 281.3 ....... +2.5 km
>
> This agrees to ~1 km with the results that Peter reported. Since the
> object of the "burn" was to raise perigee (with very little change
> in apogee), and since a post-burn mean motion is not yet available,
> I haven't bothered to examine the apogee differences.
>
> 73 de Tom, W3IWI
> Date: Mon, 25 Jun 2001 22:24:29 +0900
> From: Mineo Wakita
> To: tac
> Subject: Re: AO-40: more successful ARCJET operations
>
> Thank you, Tom Clark / W3IWI.
>
> I confirmed the equatorial radius of the oblate earth is 6378.140 km
> (6378140 m = 637814000 cm) by a astronomical journal of Japanese.
> As a result, this perigee height agrees to ~1 km with the results
> that Peter reported.
>
> I corrected as the following (18),(19) and I reposted the Excel sheet.
> http://www.ne.jp/asahi/hamradio/je9pel/satrange.zip
>
>
> (18) r0=a(1-e^2)/(1+e*cos(s)) : Distance from the center of the earth
> (19) r=r0-637814000 : Distance from the surface of the earth
>
>
> The solution of this algorithm corrected by W3IWI and JE9PEL :
>
> Before After
> Perigee height : 278.8 281.3 ....... +2.5 km
> Apogee height : 58960.0 58959.7 ....... -0.3 km
>
>
> The report of Peter Guelzow, DB2OS :
>
> Before After
> Perigee height : 279.754 282.2427 .... +2.5 km
> Apogee height : 58971.166 58971.024 .... -0.1 km
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