A. B. Carleial Consulting

DIVERTIMENTO 1

You can track the first Brazilian satellite, SCD1, on the internet in real time. Go to http://www.n2yo.com/?s=22490 and follow the spacacraft as it moves along its nearly circular low-inclination orbit around the Earth at an average altitude of about 750km.

You can likewise track SCD2 at http://www.n2yo.com/?s=25504. The two satellites have very similar orbital parameters, including an orbital period of almost exactly 100 minutes.

SCD1 was launched on February 9, 1993, and SCD2 was launched on October 23, 1998. Both satellites were designed, developed, and built in Brazil, and both remain in operation to this date, relaying environmental data collected on the ground by automatic DCPs to tracking stations located in Cuiabá, MT, and Alcântara, MA. 

How about tracking a remote-sensing satellite in a high-inclination (polar) orbit? The fourth Chinese-Brazilian Earth Resources Satellite, CBERS4, can be followed at http://www.n2yo.com/?s=40336. The most recent satellite of this binational series, CBERS4A, launched in December, 2019, is found at https://heavens-above.com/orbit.aspx?satid=44883&lat=0&lng=0&loc=Unspecified&alt=0&tz=UCT.

And here is the link for Amazonia-1 http://www.n2yo.com/?s=47699 , the first entirely Brazilian remote sensing satellite, launched on February 28, 2021. Notwithstanding its name, Amazonia-1 is also in a polar orbit and can make images of any place on Earth.

Shown in the photo below is part of a model of one of the first satellites of the CBERS Program assembled for tests at INPE's Integration and Testing Laboratory (LIT) in São José dos Campos.

Você pode acompanhar em tempo real os dois satélites brasileiros de coleta de dados ambientais em suas trajetórias ao redor da Terra clicando em http://www.n2yo.com/?s=22490 para o SCD1 e em http://www.n2yo.com/?s=25504 para o SCD2. Ambos foram inteiramente projetados, desenvolvidos, construídos e testados no Brasil há mais de trinta anos. Lançados em 1993 e 1998, respectivamente, continuam funcionando em órbita até hoje. Como a inclinação das órbitas desses satélites em relação ao plano do equador é de 25 graus, eles não sobrevoam locais com latitude maior que 25 graus; mas podem "enxergar" um pouco mais além, graças à altitude das órbitas. Costumo dizer que os SCDs são "satélites tropicais".

Da mesma forma, você pode acompanhar em http://www.n2yo.com/?s=40336 a trajetória do quarto satélite sino-brasileiro de observação da Terra (sensoreamento remoto) CBERS4, de órbita quase-polar, lançado em dezembro de 2014, que sobrevoa pontos desde o equador até as vizinhanças dos polos (sua cobertura é global). O mais recente satélite desta série, o CBERS4A, lançado em dezembro de 2019, encontra-se em https://heavens-above.com/orbit.aspx?satid=44883&lat=0&lng=0&loc=Unspecified&alt=0&tz=UCT  .

E aqui está o Amazônia-1 http://www.n2yo.com/?s=47699 , o primeiro satélite de sensoreamento remoto inteiramente brasileiro, lançado a 28 de fevereiro de 2021. A despeito do seu nome, o Amazônia-1 também está em órbita polar e pode imagear qualquer ponto da Terra.

A foto acima mostra parte de um modelo de um dos primeiros satélites do Programa CBERS montado para testes no Laboratório de Integração e Testes (LIT) do INPE em São José dos Campos.



DIVERTIMENTO 3

This is a mathematical divertissement with prime numbers.

The following are the smallest prime numbers that, when written in binary digits, are made up of 1s only (no 0s):

11 (=3)   111 (=7)   11111 (=31)   1111111 (=127)

These numbers are known as Mersenne primes. Clearly, each is a power of 2 minus 1:  3=4-1,  7=8-1,  31=32-1, and so on. By the way, most numbers of this form, such as 1111 (=15), are not prime.

It is conjectured, but has never been proved, that the sequence of all Mersenne primes is infinite. In other words, people believe that there is no largest prime number of this form.

Now consider the infinite sequence of all odd numbers that are written in binary with alternating 0s and 1s. It begins as follows:

1 (=1)   101 (=5)   10101 (=21)   1010101 (=85)   101010101 (=341)  

10101010101 (=1365)   1010101010101 (=5461)

101010101010101 (=21845)   10101010101010101 (=87381) ...

None of the nine numbers above, except 101, is prime. (As you know, 1 is not considered to be prime, it is hors concours...)

Is 101 (=5) the only prime number in this infinite sequence? If so, can you prove it? If not, what is the next prime in the sequence?

If you are convinced that 5 is not alone, is there a largest prime number (larger than 5) that is made up of alternating 0s and 1s? Or is there an interminable (infinite) list of primes made up of alternating 0s and 1s?

Note: The above questions were first posted here more than five years ago, and I have not yet received a correct answer. Nobody has sent me an example of a prime number of the 1010...10101 class that is greater than five. (Hint: It would have to be a huge number, a number with more than one hundred decimal digits.) And nobody has sent me a valid proof that no such prime number exists. Is the answer known, or do we have an open problem in mathematics?

DIVERTIMENTO 2

A geostationary satellite appears fixed in the sky because it goes around the Earth on a circular equatorial orbit in exactly one sidereal day (T=23h56min4.1sec). The radius of that unique orbit is R=42164km, the constant speed is V=3074.6m/s, and the centripetal acceleration (V2/R) caused by gravity equals GM/R2, where G is Newton's constant and M is the mass of the Earth.

Similarly, a satellite of Mars that will appear fixed in the sky to an observer on the surface of that planet must be placed in equatorial orbit at a precise altitude and velocity. Can you determine the radius of the unique ares-stationary satellite orbit? As you know, the Martian sidereal day is slightly longer than ours (24h37min22.7sec) and the mass of Mars is rather small, only 0.10745 of the mass of the Earth.

Indeed, it should be possible to launch and stationkeep and operate stationary satellites around Mars! Yet, after you compute their altitude, you may wonder if gravitational perturbations by Deimos and Phobos will be a serious concern. Should that be a cause for fear of spacecraft loss? 

On the same subject, do you think it is possible to place stationary satellites around Mercury, Venus, or our Moon? Why not?

Um satélite geoestacionário aparece "parado" no céu porque dá uma volta em torno da Terra em órbita circular equatorial em exatamente um dia sideral (T=86164,1seg). O raio da órbita é R=42164km.

Um satélite aresestacionário fará a mesma coisa em órbita em torno do planeta Marte (cujo nome em grego é Ares).

Exercício: Sabendo que o dia sideral de Marte é um pouquinho mais longo (T=88642,7seg) e a massa de Marte é apenas 10,745% da massa da Terra, calcule o raio da órbita dos futuros satélites aresestacionários.

Dica: Há proporcionalidade envolvendo as massas, os quadrados dos períodos e os cubos dos raios das órbitas (João Kepler).